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currentKernel; ] :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] Math 325: Differential Equations Solutions to Assignment 3 :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Initialization :[font = input; initialization; noKeepOnOnePage; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450; endGroup] *) <>"] ;[o] -Graphics- :[font = text; inactive; preserveAspect] The solutions are unstable spirals that flow away from the origin. :[font = text; inactive; preserveAspect] b) ;[s] 1:0,0;3,-1; 1:1,13,9,Times,2,12,0,0,0; :[font = input; preserveAspect] A = {{-3, 0, 2}, { 1,-1, 0}, {-2,-1, 0}}; :[font = text; inactive; preserveAspect] Get the eigenvalues and eigenvectors and put them in variables r[i] and v[i], respectively. ;[s] 5:0,0;63,1;67,0;72,1;76,0;92,-1; 2:3,13,9,Times,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] es = Eigensystem[A]//Simplify :[font = output; output; inactive; preserveAspect; endGroup] {{-2, -1 - I*2^(1/2), -1 + I*2^(1/2)}, {{2, -2, 1}, {(2*I)/(2*I + 2^(1/2)), -(2^(1/2)/(2*I + 2^(1/2))), 1}, {(-2*I)/(-2*I + 2^(1/2)), -(2^(1/2)/(-2*I + 2^(1/2))), 1}}} ;[o] {{-2, -1 - I Sqrt[2], -1 + I Sqrt[2]}, 2 I Sqrt[2] {{2, -2, 1}, {-------------, -(-------------), 1}, 2 I + Sqrt[2] 2 I + Sqrt[2] -2 I Sqrt[2] {--------------, -(--------------), 1}}} -2 I + Sqrt[2] -2 I + Sqrt[2] :[font = input; preserveAspect] Do[r[i] = es[[1,i]]; v[i] = es[[2,i]];, {i,1,3}] :[font = text; inactive; preserveAspect] The first eigenvalue gives a real-valued solution: :[font = input; preserveAspect; startGroup] X1[t_] = v[1]E^(r[1]t) :[font = output; output; inactive; preserveAspect; endGroup] {2/E^(2*t), -2/E^(2*t), E^(-2*t)} ;[o] 2 -2 -2 t {----, ----, E } 2 t 2 t E E :[font = text; inactive; preserveAspect] The other two eigenevalues are complex conjugates. We obtain two real-valued solutions from the real and imaginary parts of the complex-valued solution. :[font = input; preserveAspect; startGroup] Z[t_] = v[3]E^(r[3]t) :[font = output; output; inactive; preserveAspect; endGroup] {(-2*I*E^((-1 + I*2^(1/2))*t))/(-2*I + 2^(1/2)), -((2^(1/2)*E^((-1 + I*2^(1/2))*t))/(-2*I + 2^(1/2))), E^((-1 + I*2^(1/2))*t)} ;[o] (-1 + I Sqrt[2]) t (-1 + I Sqrt[2]) t -2 I E Sqrt[2] E {------------------------, -(---------------------------), -2 I + Sqrt[2] -2 I + Sqrt[2] (-1 + I Sqrt[2]) t E } :[font = input; preserveAspect; startGroup] X2[t_] = Re[Z[t]]//ComplexExpand :[font = output; output; inactive; preserveAspect; endGroup] {(2*Cos[2^(1/2)*t])/(3*E^t) + (2^(1/2)*Sin[2^(1/2)*t])/(3*E^t), -Cos[2^(1/2)*t]/(3*E^t) + (2^(1/2)*Sin[2^(1/2)*t])/(3*E^t), Cos[2^(1/2)*t]/E^t} ;[o] 2 Cos[Sqrt[2] t] Sqrt[2] Sin[Sqrt[2] t] {---------------- + ----------------------, t t 3 E 3 E -Cos[Sqrt[2] t] Sqrt[2] Sin[Sqrt[2] t] Cos[Sqrt[2] t] --------------- + ----------------------, --------------} t t t 3 E 3 E E :[font = input; preserveAspect; startGroup] X3[t_] = Im[Z[t]]//ComplexExpand :[font = output; output; inactive; preserveAspect; endGroup] {-(2^(1/2)*Cos[2^(1/2)*t])/(3*E^t) + (2*Sin[2^(1/2)*t])/(3*E^t), -(2^(1/2)*Cos[2^(1/2)*t])/(3*E^t) - Sin[2^(1/2)*t]/(3*E^t), Sin[2^(1/2)*t]/E^t} ;[o] -(Sqrt[2] Cos[Sqrt[2] t]) 2 Sin[Sqrt[2] t] {------------------------- + ----------------, t t 3 E 3 E -(Sqrt[2] Cos[Sqrt[2] t]) Sin[Sqrt[2] t] Sin[Sqrt[2] t] ------------------------- - --------------, --------------} t t t 3 E 3 E E :[font = text; inactive; preserveAspect] The general solution is :[font = input; preserveAspect; startGroup] X[t_] = c1 X1[t] + c2 X2[t] + c3 X3[t] :[font = output; output; inactive; preserveAspect; endGroup] {(2*c1)/E^(2*t) + c3*(-(2^(1/2)*Cos[2^(1/2)*t])/(3*E^t) + (2*Sin[2^(1/2)*t])/(3*E^t)) + c2*((2*Cos[2^(1/2)*t])/(3*E^t) + (2^(1/2)*Sin[2^(1/2)*t])/(3*E^t)), (-2*c1)/E^(2*t) + c3*(-(2^(1/2)*Cos[2^(1/2)*t])/(3*E^t) - Sin[2^(1/2)*t]/(3*E^t)) + c2*(-Cos[2^(1/2)*t]/(3*E^t) + (2^(1/2)*Sin[2^(1/2)*t])/(3*E^t)) , c1/E^(2*t) + (c2*Cos[2^(1/2)*t])/E^t + (c3*Sin[2^(1/2)*t])/E^t} ;[o] 2 c1 -(Sqrt[2] Cos[Sqrt[2] t]) 2 Sin[Sqrt[2] t] {---- + c3 (------------------------- + ----------------) + 2 t t t E 3 E 3 E 2 Cos[Sqrt[2] t] Sqrt[2] Sin[Sqrt[2] t] c2 (---------------- + ----------------------), t t 3 E 3 E -2 c1 -(Sqrt[2] Cos[Sqrt[2] t]) Sin[Sqrt[2] t] ----- + c3 (------------------------- - --------------) + 2 t t t E 3 E 3 E -Cos[Sqrt[2] t] Sqrt[2] Sin[Sqrt[2] t] c2 (--------------- + ----------------------), t t 3 E 3 E c1 c2 Cos[Sqrt[2] t] c3 Sin[Sqrt[2] t] ---- + ----------------- + -----------------} 2 t t t E E E :[font = text; inactive; preserveAspect] Let's check: :[font = input; preserveAspect; startGroup] X'[t]//Simplify :[font = output; output; inactive; preserveAspect; endGroup] {(-4*c1 + 2^(1/2)*c3*E^t*Cos[2^(1/2)*t] - 2^(1/2)*c2*E^t*Sin[2^(1/2)*t])/E^(2*t), (4*c1 + c2*E^t*Cos[2^(1/2)*t] + c3*E^t*Sin[2^(1/2)*t])/E^(2*t), (-2*c1 - c2*E^t*Cos[2^(1/2)*t] + 2^(1/2)*c3*E^t*Cos[2^(1/2)*t] - 2^(1/2)*c2*E^t*Sin[2^(1/2)*t] - c3*E^t*Sin[2^(1/2)*t])/ E^(2*t)} ;[o] t {(-4 c1 + Sqrt[2] c3 E Cos[Sqrt[2] t] - t 2 t Sqrt[2] c2 E Sin[Sqrt[2] t]) / E , t t 4 c1 + c2 E Cos[Sqrt[2] t] + c3 E Sin[Sqrt[2] t] --------------------------------------------------, 2 t E t t (-2 c1 - c2 E Cos[Sqrt[2] t] + Sqrt[2] c3 E Cos[Sqrt[2] t] - t t 2 t Sqrt[2] c2 E Sin[Sqrt[2] t] - c3 E Sin[Sqrt[2] t]) / E } :[font = input; preserveAspect; startGroup] A.X[t]//Simplify :[font = output; output; inactive; preserveAspect; endGroup] {(-4*c1 + 2^(1/2)*c3*E^t*Cos[2^(1/2)*t] - 2^(1/2)*c2*E^t*Sin[2^(1/2)*t])/E^(2*t), (4*c1 + c2*E^t*Cos[2^(1/2)*t] + c3*E^t*Sin[2^(1/2)*t])/E^(2*t), (-2*c1 - c2*E^t*Cos[2^(1/2)*t] + 2^(1/2)*c3*E^t*Cos[2^(1/2)*t] - 2^(1/2)*c2*E^t*Sin[2^(1/2)*t] - c3*E^t*Sin[2^(1/2)*t])/E^(2*t) } ;[o] t {(-4 c1 + Sqrt[2] c3 E Cos[Sqrt[2] t] - t 2 t Sqrt[2] c2 E Sin[Sqrt[2] t]) / E , t t 4 c1 + c2 E Cos[Sqrt[2] t] + c3 E Sin[Sqrt[2] t] --------------------------------------------------, 2 t E t t (-2 c1 - c2 E Cos[Sqrt[2] t] + Sqrt[2] c3 E Cos[Sqrt[2] t] - t t 2 t Sqrt[2] c2 E Sin[Sqrt[2] t] - c3 E Sin[Sqrt[2] t]) / E } :[font = text; inactive; preserveAspect] Now we plot several solutions. :[font = input; preserveAspect] solns = Flatten[Table[X[t],{c1,-1,1},{c2,-1,1},{c3,-1,1}],2]; :[font = input; preserveAspect; startGroup] p = ParametricPlot3D[Evaluate[solns], {t,-Pi,Pi}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 281; pictureHeight = 270] %! %%Creator: Mathematica %%AspectRatio: .96105 MathPictureStart %% Graphics3D /Courier findfont 10 scalefont setfont % Scaling calculations 0.022757 0.98458 0.0246145 0.98458 [ [(-10)] .19656 .21237 .93577 1 Msboxa [(0)] .36665 .14449 .82487 1 Msboxa [(10)] .54849 .07196 .71396 1 Msboxa [(-10)] .08221 .74264 .12526 -1 Msboxa [(0)] .20445 .83434 .08756 -1 Msboxa 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s P p .004 w .50803 .50091 m .5083 .50386 L s P p .004 w .51078 .52364 m .51352 .52428 L s P p .004 w .53837 .51365 m .53521 .51362 L s P p .004 w .49422 .51957 m .49931 .52171 L s P p .004 w .52696 .51942 m .52649 .51953 L s P p .004 w .53555 .50548 m .53183 .50643 L s P p .004 w .54366 .53406 m .5423 .5318 L s P p .004 w .5196 .52675 m .52135 .52663 L s P p .004 w .50812 .49754 m .50803 .50091 L s P p .004 w .51852 .51364 m .51881 .51487 L s P p .004 w .52272 .49174 m .51949 .49487 L s P p .004 w .51743 .52666 m .5196 .52675 L s P p .004 w .49931 .52171 m .50392 .52342 L s P p .004 w .52486 .50722 m .52331 .50875 L s P p .004 w .53919 .53363 m .53866 .53179 L s P p .004 w .5315 .51572 m .52996 .51607 L s P p .004 w .6693 .57385 m .65091 .56011 L s P p .004 w .51479 .52632 m .51743 .52666 L s P p .004 w .52733 .51913 m .52696 .51942 L s P p .004 w .50865 .49374 m .50812 .49754 L s P p .004 w .50392 .52342 m .50805 .52474 L s P p .004 w .51167 .52569 m .51479 .52632 L s P p .004 w 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P p .004 w .54379 .54281 m .54478 .54079 L s P p .004 w .51847 .4983 m .518 .5021 L s P p .004 w .53797 .47969 m .5318 .48424 L s P p .004 w .55001 .50122 m .54457 .50294 L s P p .004 w .20365 .45591 m .24632 .48368 L s P p .004 w .55793 .48656 m .54933 .48941 L s P p .004 w .5851 .55686 m .58096 .55112 L s P p .004 w .53468 .49752 m .53154 .50062 L s P p .004 w .53275 .54089 m .53558 .53991 L s P p .004 w .52766 .51318 m .52778 .51519 L s P p .004 w .53848 .51171 m .53659 .51325 L s P p .004 w .6879 .58918 m .6693 .57385 L s P p .004 w .45676 .52074 m .46989 .52662 L s P p .004 w .54191 .54459 m .54379 .54281 L s P p .004 w .51934 .49383 m .51847 .4983 L s P p .004 w .52326 .46582 m .51812 .47286 L s P p .004 w .59925 .56257 m .59354 .55556 L s P p .004 w .52896 .54147 m .53275 .54089 L s P p .004 w .55367 .512 m .54942 .51282 L s P p .004 w .5275 .5106 m .52766 .51318 L s P p .004 w .54549 .47447 m .53797 .47969 L s P p .004 w .53902 .54603 m .54191 .54459 L s P p .004 w .55616 .49906 m .55001 .50122 L s P p .004 w .43498 .51784 m .45173 .52578 L s P p .004 w .46989 .52662 m .48196 .53142 L s P p .004 w .52411 .54157 m .52896 .54147 L s P p .004 w .53843 .49382 m .53468 .49752 L s P p .004 w .54052 .50967 m .53848 .51171 L s P p .004 w .52074 .48858 m .51934 .49383 L s P p .004 w .51813 .54109 m .52411 .54157 L s P p .004 w .58836 .56277 m .5851 .55686 L s P p .004 w .53 .4578 m .52326 .46582 L s P p .004 w .48196 .53142 m .49286 .53518 L s P p .004 w .56793 .48319 m .55793 .48656 L s P p .004 w .52734 .50734 m .5275 .5106 L s P p .004 w .535 .54705 m .53902 .54603 L s P p .004 w .51095 .53993 m .51813 .54109 L s P p .004 w .49286 .53518 m .50253 .53799 L s P p .004 w .55823 .51075 m .55367 .512 L s P p .004 w .50253 .53799 m .51095 .53993 L s P p .004 w .52284 .48243 m .52074 .48858 L s P p .004 w .60405 .56988 m .59925 .56257 L s P p .004 w .45173 .52578 m .46729 .53235 L s P p .004 w .54274 .50702 m .54052 .50967 L s P p .004 w .52971 .54752 m .535 .54705 L s P p .004 w .54294 .48937 m .53843 .49382 L s P p .004 w .52725 .50329 m .52734 .50734 L s P p .004 w .55461 .4684 m .54549 .47447 L s P p .004 w .56315 .49634 m .55616 .49906 L s P p .004 w .52307 .54734 m .52971 .54752 L s P p .004 w .46729 .53235 m .48148 .5376 L s P p .004 w .59047 .56875 m .58836 .56277 L s P p .004 w .53868 .44862 m .53 .4578 L s P p .004 w .52582 .47522 m .52284 .48243 L s P p .004 w .51497 .54638 m .52307 .54734 L s P p .004 w .74096 .61636 m .71709 .59608 L s P p .004 w .56316 .50893 m .55823 .51075 L s P p .004 w .70632 .60609 m .6879 .58918 L s P p .004 w .54522 .50361 m .54274 .50702 L s P p .004 w .48148 .5376 m .49419 .54162 L s P p .004 w .52732 .49831 m .52725 .50329 L s P p .004 w .57955 .47914 m .56793 .48319 L s P p .004 w .50535 .54451 m .51497 .54638 L s P p .004 w .49419 .54162 m .50535 .54451 L s P p .004 w .54835 .48402 m .54294 .48937 L s P p .004 w .60764 .57737 m .60405 .56988 L s P p .004 w .57109 .4929 m .56315 .49634 L s P p .004 w .52993 .46679 m .52582 .47522 L s P p .004 w .56563 .46129 m .55461 .4684 L s P p .004 w .59114 .57465 m .59047 .56875 L s P p .004 w .52766 .49223 m .52732 .49831 L s P p .004 w .54972 .43803 m .53868 .44862 L s P p .004 w .54802 .49928 m .54522 .50361 L s P p .004 w .24632 .48368 m .28951 .50879 L s P p .004 w .5685 .50637 m .56316 .50893 L s P p .004 w .55487 .47759 m .54835 .48402 L s P p .004 w .60966 .58491 m .60764 .57737 L s P p .004 w .52844 .48487 m .52766 .49223 L s P p .004 w .59306 .47419 m .57955 .47914 L s P p .004 w .53546 .45689 m .52993 .46679 L s P p .004 w .59005 .58031 m .59114 .57465 L s P p .004 w .58015 .48853 m .57109 .4929 L s P p .004 w .55128 .49383 m .54802 .49928 L s P p .004 w .57892 .45287 m .56563 .46129 L s P p .004 w .72405 .62457 m .70632 .60609 L s P p .004 w .57433 .50287 m .5685 .50637 L s P p .004 w .56363 .42571 m .54972 .43803 L s P p .004 w .52986 .47601 m .52844 .48487 L s P p .004 w .76463 .63884 m .74096 .61636 L s P p .004 w .56275 .4698 m .55487 .47759 L s P p .004 w .60972 .59229 m .60966 .58491 L s P p .004 w .54279 .44523 m .53546 .45689 L s P p .004 w .58686 .58552 m .59005 .58031 L s P p .004 w .55514 .48702 m .55128 .49383 L s P p .004 w .60881 .46805 m .59306 .47419 L s P p .004 w .59054 .48295 m .58015 .48853 L s P p .004 w .53216 .46538 m .52986 .47601 L s P p .004 w .28951 .50879 m .332 .53085 L s P p .004 w .58076 .49818 m .57433 .50287 L s P p .004 w .58119 .59005 m .58686 .58552 L s P p .004 w .59498 .44277 m .57892 .45287 L s P p .004 w .57232 .46036 m .56275 .4698 L s P p .004 w .60738 .59929 m .60972 .59229 L s P p .004 w .58107 .41122 m .56363 .42571 L s P p .004 w .55237 .43146 m .54279 .44523 L s P p .004 w .5598 .47855 m .55514 .48702 L s P p .004 w .53567 .45264 m .53216 .46538 L s P p .004 w .74049 .64453 m .72405 .62457 L s P p .004 w .57269 .59364 m .58119 .59005 L s P p .004 w .60255 .47584 m .59054 .48295 L s P p .004 w .58795 .49197 m .58076 .49818 L s P p .004 w .60219 .60563 m .60738 .59929 L s P p .004 w .78747 .66351 m .76463 .63884 L s P p .004 w .332 .53085 m .37274 .54964 L s P p .004 w .62725 .46033 m .60881 .46805 L s P p .004 w .58398 .44884 m .57232 .46036 L s P p .004 w .561 .59598 m .57269 .59364 L s P p .004 w .56553 .46806 m .5598 .47855 L s P p .004 w .56479 .41509 m .55237 .43146 L s P p .004 w .61443 .43052 m .59498 .44277 L s P p .004 w .5408 .43737 m .53567 .45264 L s P p .004 w .60291 .39395 m .58107 .41122 L s P p .004 w .59366 .61098 m .60219 .60563 L s P p .004 w .54578 .59674 m .561 .59598 L s P p .004 w .37274 .54964 m .41084 .56506 L s P p .004 w .26127 .52218 m .31287 .54775 L s P p .004 w .59611 .48385 m .58795 .49197 L s P p .004 w .75361 .66399 m .74049 .64453 L s P p .004 w .61654 .46673 m .60255 .47584 L s P p .004 w .52677 .59559 m .54578 .59674 L s P p .004 w .57268 .45508 m .56553 .46806 L s P p .004 w .54808 .41906 m .5408 .43737 L s P p .004 w .59829 .43472 m .58398 .44884 L s P p .004 w .41084 .56506 m .44565 .57717 L s P p .004 w .58132 .61499 m .59366 .61098 L s P p .004 w .58082 .39546 m .56479 .41509 L s P p .004 w .64899 .45052 m .62725 .46033 L s P p .004 w .50378 .59216 m .52677 .59559 L s P p .004 w .44565 .57717 m .47671 .58612 L s P p .004 w .47671 .58612 m .50378 .59216 L s P p .004 w .79296 .67047 m .78747 .66351 L s P p .004 w .6381 .41547 m .61443 .43052 L s P p .004 w .63029 .37309 m .60291 .39395 L s P p .004 w .56471 .61724 m .58132 .61499 L s P p .004 w .31287 .54775 m .36209 .56921 L s P p .004 w .60553 .47332 m .59611 .48385 L s P p .004 w .58172 .43902 m .57268 .45508 L s P p .004 w .5582 .397 m .54808 .41906 L s P p .004 w .63302 .45506 m .61654 .46673 L s P p .004 w .54342 .61728 m .56471 .61724 L s P p .004 w .61596 .41727 m .59829 .43472 L s P p .004 w .36209 .56921 m .40781 .58648 L s P p .004 w .60147 .3717 m .58082 .39546 L s P p .004 w .51715 .61465 m .54342 .61728 L s P p .004 w .67484 .43792 m .64899 .45052 L s P p .004 w .40781 .58648 m .44921 .59964 L s P p .004 w .48574 .60891 m .51715 .61465 L s P p .004 w .61661 .45972 m .60553 .47332 L s P p .004 w .66716 .39671 m .6381 .41547 L s P p .004 w .44921 .59964 m .48574 .60891 L s P p .004 w .59328 .41911 m .58172 .43902 L s P p .004 w .57205 .37027 m .5582 .397 L s P p .004 w .66482 .34745 m .63029 .37309 L s P p .004 w .65266 .44004 m .63302 .45506 L s P p .004 w .63798 .39553 m .61596 .41727 L s P p .004 w .62602 .34488 m .60147 .3717 L s P p .004 w .62993 .44221 m .61661 .45972 L s P p .004 w .70594 .42155 m .67484 .43792 L s P p .004 w .60819 .39432 m .59328 .41911 L s P p .004 w .57733 .3611 m .57205 .37027 L s P p .004 w .70319 .37295 m .66716 .39671 L s P p .004 w .67642 .42061 m .65266 .44004 L s P p .004 w .68562 .33224 m .66482 .34745 L s P p .004 w .65492 .37881 m .63798 .39553 L s P p .004 w .64321 .42385 m .62993 .44221 L s P p .004 w .60904 .39296 m .60819 .39432 L s P p .004 w .73124 .4072 m .70594 .42155 L s P p .004 w .71093 .36773 m .70319 .37295 L s P p .004 w .68684 .41158 m .67642 .42061 L s P p .002 w .69995 .02461 m .92199 .37664 L s .92199 .37664 m .97539 .78197 L s .97539 .78197 m .7295 .46199 L s .7295 .46199 m .69995 .02461 L s .08476 .26914 m .02461 .68594 L s .02461 .68594 m .7295 .46199 L s .7295 .46199 m .69995 .02461 L s .69995 .02461 m .08476 .26914 L s P p p .002 w .08476 .26914 m .69995 .02461 L s P p .002 w .20497 .22136 m .20917 .22585 L s P [(-10)] .19656 .21237 .93577 1 Mshowa p .002 w .37448 .15398 m .3784 .15873 L s P [(0)] .36665 .14449 .82487 1 Mshowa p .002 w .55564 .08198 m .55921 .08698 L s P [(10)] .54849 .07196 .71396 1 Mshowa p .001 w .23799 .20823 m .24048 .21096 L s P p .001 w .27145 .19493 m .27391 .19769 L s P p .001 w .30534 .18146 m .30777 .18425 L s P p .001 w .33968 .16781 m .34207 .17063 L s P p .001 w .40974 .13996 m .41206 .14284 L s P p .001 w .44548 .12576 m .44775 .12867 L s P p .001 w .4817 .11136 m .48393 .11431 L s P p .001 w .51842 .09677 m .52061 .09974 L s P p .001 w .17236 .23432 m .17492 .23698 L s P p .001 w .14017 .24711 m .14276 .24975 L s P p .001 w .10839 .25975 m .111 .26235 L s P p .001 w .59337 .06698 m .59547 .07001 L s P p .001 w .63163 .05177 m .63369 .05484 L s P p .001 w .67043 .03635 m .67244 .03945 L s P P p p .002 w .02461 .68594 m .35747 .93643 L s P p .002 w .08374 .73043 m .0845 .72432 L s P [(-10)] .08221 .74264 .12526 -1 Mshowa p .002 w .20552 .82208 m .20606 .81595 L s P [(0)] .20445 .83434 .08756 -1 Mshowa p .002 w .31346 .90331 m .3138 .89717 L s P [(10)] .31277 .9156 .05575 -1 Mshowa p .001 w .10933 .74969 m .10976 .74602 L s P p .001 w .13428 .76847 m .13468 .7648 L s P p .001 w .15861 .78678 m .15899 .78311 L s P p .001 w .18235 .80465 m .1827 .80097 L s P p .001 w .22814 .8391 m .22843 .83542 L s P p .001 w .25022 .85572 m .25049 .85204 L s P p .001 w .27179 .87195 m .27204 .86827 L s P p .001 w .29286 .88781 m .29309 .88413 L s P p .001 w .05748 .71067 m .05797 .70701 L s P p .001 w .03053 .69039 m .03105 .68674 L s P p .001 w .33359 .91846 m .33377 .91477 L s P p .001 w .35327 .93328 m .35344 .92959 L s P P p P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics3D["<<>>"] ;[o] -Graphics3D- :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Problem 2 :[font = text; inactive; preserveAspect; endGroup] Using eigenvalues and eigenvectors, find the general solution of the system X'[t] == A.X[t] for each of the matrices A below. Each system has a repeated eigenvalue and generalized eigenevectors must be used. Check that your solution is correct by comparing X'[t] and A.X[t]. Then plot several solutions for each using ParametricPlot[] or ParametricPlot3D[]. a) A = {{ 6,-4}, { 9,-6}}; b) A = {{-1, 1 -1}, { 1,-3, 1}, { 2, 0,-2}}; ;[s] 21:0,0;76,2;95,0;121,2;122,0;261,2;266,0;271,2;277,0;322,2;338,0;342,2;360,0;363,1;366,0;367,2;398,0;399,1;401,0;403,2;458,0;460,-1; 3:11,13,9,Times,0,12,0,0,0;2,13,9,Times,2,12,0,0,0;8,13,10,Courier,1,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Solution :[font = text; inactive; preserveAspect] a) ;[s] 1:0,0;3,-1; 1:1,13,9,Times,2,12,0,0,0; :[font = input; preserveAspect] A = {{ 6, -4}, { 9, -6}}; :[font = input; preserveAspect; startGroup] Eigensystem[A] :[font = output; output; inactive; preserveAspect; endGroup] {{0, 0}, {{2, 3}, {0, 0}}} ;[o] {{0, 0}, {{2, 3}, {0, 0}}} :[font = text; inactive; preserveAspect] The repeated eigenvalue is 0, and Eigensystem[] finds only one non-trivial eigenvector, {2,3}. This vector also gives a (constant) fundamental solution ;[s] 7:0,0;27,1;28,0;34,1;47,0;88,1;93,0;152,-1; 2:4,13,9,Times,0,12,0,0,0;3,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] X1[t_] = {2,3}E^(0*t) :[font = output; output; inactive; preserveAspect; endGroup] {2, 3} ;[o] {2, 3} :[font = text; inactive; preserveAspect] A generalized eigenvector is found by solving :[font = input; preserveAspect; startGroup] Solve[(A - 0*IdentityMatrix[2]).{w1,w2} == {2,3}] :[font = message; inactive; preserveAspect] Solve::svars: Warning: Equations may not give solutions for all "solve" variables. :[font = output; output; inactive; preserveAspect; endGroup] {{w1 -> 1/3 + (2*w2)/3}} ;[o] 1 2 w2 {{w1 -> - + ----}} 3 3 :[font = text; inactive; preserveAspect] So {1/3, 0} is one generalized eigenvector giving us a second fundamental solution: ;[s] 3:0,0;3,1;11,0;84,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] X2[t_] = {2,3} t E^(0*t) + {1/3, 0} E^(0*t) :[font = output; output; inactive; preserveAspect; endGroup] {1/3 + 2*t, 3*t} ;[o] 1 {- + 2 t, 3 t} 3 :[font = text; inactive; preserveAspect] The general solution is: :[font = input; preserveAspect; startGroup] X[t_] = c1 X1[t] + c2 X2[t] :[font = output; output; inactive; preserveAspect; endGroup] {2*c1 + c2*(1/3 + 2*t), 3*c1 + 3*c2*t} ;[o] 1 {2 c1 + c2 (- + 2 t), 3 c1 + 3 c2 t} 3 :[font = text; inactive; preserveAspect] Let's check: :[font = input; preserveAspect; startGroup] X'[t] :[font = output; output; inactive; preserveAspect; endGroup] {2*c2, 3*c2} ;[o] {2 c2, 3 c2} :[font = input; preserveAspect; startGroup] A.X[t]//Simplify :[font = output; output; inactive; preserveAspect; endGroup] {2*c2, 3*c2} ;[o] {2 c2, 3 c2} :[font = text; inactive; preserveAspect] Multiples of the first fundamental solution give a line of fixed points: :[font = input; preserveAspect; startGroup] p1 = ParametricPlot[Evaluate[c1 X1[t]], {c1,-2,2}, PlotStyle->RGBColor[1,0,0]] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.119048 0.309017 0.0490503 [ [(-4)] .02381 .30902 0 2 Msboxa [(-2)] .2619 .30902 0 2 Msboxa [(2)] .7381 .30902 0 2 Msboxa [(4)] .97619 .30902 0 2 Msboxa [(-6)] .4875 .01472 1 0 Msboxa [(-4)] .4875 .11282 1 0 Msboxa [(-2)] .4875 .21092 1 0 Msboxa [(2)] .4875 .40712 1 0 Msboxa [(4)] .4875 .50522 1 0 Msboxa [(6)] .4875 .60332 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .02381 .30902 m .02381 .31527 L s P [(-4)] .02381 .30902 0 2 Mshowa p .002 w .2619 .30902 m .2619 .31527 L s P [(-2)] .2619 .30902 0 2 Mshowa p .002 w .7381 .30902 m .7381 .31527 L s P [(2)] .7381 .30902 0 2 Mshowa p .002 w .97619 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preserveAspect; startGroup] p2 = ParametricPlot[Evaluate[X2[t]], {t,-2,2}, PlotStyle->RGBColor[0,0,1]] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.460317 0.119048 0.309017 0.0490503 [ [(-2)] .22222 .30902 0 2 Msboxa [(2)] .69841 .30902 0 2 Msboxa [(4)] .93651 .30902 0 2 Msboxa [(-6)] .44782 .01472 1 0 Msboxa [(-4)] .44782 .11282 1 0 Msboxa [(-2)] .44782 .21092 1 0 Msboxa [(2)] .44782 .40712 1 0 Msboxa [(4)] .44782 .50522 1 0 Msboxa [(6)] .44782 .60332 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .22222 .30902 m .22222 .31527 L s P [(-2)] .22222 .30902 0 2 Mshowa p .002 w .69841 .30902 m .69841 .31527 L s P [(2)] .69841 .30902 0 2 Mshowa p .002 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endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect] solns = Flatten[Table[X[t],{c1,-3,3}, {c2,-3,3}],1]; :[font = input; preserveAspect; startGroup] p = ParametricPlot[Evaluate[solns], {t,-2,2}, PlotRange->{{-5,5},{-5,5}}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.1 0.309017 0.0618034 [ [(-4)] .1 .30902 0 2 Msboxa [(-2)] .3 .30902 0 2 Msboxa [(2)] .7 .30902 0 2 Msboxa [(4)] .9 .30902 0 2 Msboxa [(-4)] .4875 .0618 1 0 Msboxa [(-2)] .4875 .18541 1 0 Msboxa [(2)] .4875 .43262 1 0 Msboxa [(4)] .4875 .55623 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .1 .30902 m .1 .31527 L s P [(-4)] .1 .30902 0 2 Mshowa p .002 w .3 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:[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = text; inactive; preserveAspect] The non-stationary trajectories are all parallel to the line of fixed points :[font = text; inactive; preserveAspect] b) ;[s] 1:0,0;3,-1; 1:1,13,9,Times,2,12,0,0,0; :[font = input; preserveAspect] A = {{-1, 1,-1}, { 1,-3, 1}, { 2, 0,-2}}; :[font = input; preserveAspect; startGroup] Eigensystem[A] :[font = output; output; inactive; preserveAspect; endGroup] {{-2, -2, -2}, {{0, 1, 1}, {0, 0, 0}, {0, 0, 0}}} ;[o] {{-2, -2, -2}, {{0, 1, 1}, {0, 0, 0}, {0, 0, 0}}} :[font = text; inactive; preserveAspect] The eigenvalue -2 is repeated three times. Only one non-trivial eigenvector, {0,1,1}, is found which gives a fundamental solution: ;[s] 5:0,0;15,1;17,0;77,1;84,0;131,-1; 2:3,13,9,Times,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect] X1[t_] = {0,1,1}E^(-2t); :[font = text; inactive; preserveAspect] We now try to find two generalized eigenvectors. First solve: :[font = input; preserveAspect; startGroup] Solve[(A-(-2)IdentityMatrix[3]).{w1,w2,w3}=={0,1,1}] :[font = output; output; inactive; preserveAspect; endGroup] {{w2 -> -1/2 + w3, w1 -> 1/2}} ;[o] 1 1 {{w2 -> -(-) + w3, w1 -> -}} 2 2 :[font = text; inactive; preserveAspect] Choose w3 == 0 to get the generalized eigenvector {1/2,-1/2,0} and a corresponding fundamental solution: ;[s] 5:0,0;7,1;14,0;50,1;62,0;105,-1; 2:3,13,9,Times,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect] X2[t_] = {0,1,1}t*E^(-2t) + {1/2,-1/2,0}E^(-2t); :[font = text; inactive; preserveAspect] To find another, solve: :[font = input; preserveAspect; startGroup] Solve[(A-(-2)IdentityMatrix[3]).{u1,u2,u3}=={1/2,-1/2,0}] :[font = output; output; inactive; preserveAspect; endGroup] {{u2 -> 1/2 + u3, u1 -> 0}} ;[o] 1 {{u2 -> - + u3, u1 -> 0}} 2 :[font = text; inactive; preserveAspect] Choose u3 == 0 to get the second generalized eigenvector {0,1/2,0}. A third fundamental solution is then given by ;[s] 5:0,0;7,1;14,0;57,1;66,0;114,-1; 2:3,13,9,Times,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect] X3[t_] = {0,1,1}(t^2/2)*E^(-2t) + {1/2,-1/2,0}t*E^(-2t) + {0,1/2,0}E^(-2t); :[font = text; inactive; preserveAspect] And the general solution is: :[font = input; preserveAspect; startGroup] X[t_] = c1 X1[t] + c2 X2[t] + c3 X3[t]//Simplify :[font = output; output; inactive; preserveAspect; endGroup] {(c2 + c3*t)/(2*E^(2*t)), (2*c1 - c2 + c3 + 2*c2*t - c3*t + c3*t^2)/(2*E^(2*t)), (2*c1 + 2*c2*t + c3*t^2)/(2*E^(2*t))} ;[o] 2 c2 + c3 t 2 c1 - c2 + c3 + 2 c2 t - c3 t + c3 t {---------, --------------------------------------, 2 t 2 t 2 E 2 E 2 2 c1 + 2 c2 t + c3 t ---------------------} 2 t 2 E :[font = text; inactive; preserveAspect] Let's check: :[font = input; preserveAspect; startGroup] X'[t]//Simplify :[font = output; output; inactive; preserveAspect; endGroup] {(-2*c2 + c3 - 2*c3*t)/(2*E^(2*t)), (-4*c1 + 4*c2 - 3*c3 - 4*c2*t + 4*c3*t - 2*c3*t^2)/(2*E^(2*t)), (-2*c1 + c2 - 2*c2*t + c3*t - c3*t^2)/E^(2*t)} ;[o] -2 c2 + c3 - 2 c3 t {-------------------, 2 t 2 E 2 -4 c1 + 4 c2 - 3 c3 - 4 c2 t + 4 c3 t - 2 c3 t -----------------------------------------------, 2 t 2 E 2 -2 c1 + c2 - 2 c2 t + c3 t - c3 t ----------------------------------} 2 t E :[font = input; preserveAspect; startGroup] A.X[t]//Simplify :[font = output; output; inactive; preserveAspect; endGroup] {(-2*c2 + c3 - 2*c3*t)/(2*E^(2*t)), (-4*c1 + 4*c2 - 3*c3 - 4*c2*t + 4*c3*t - 2*c3*t^2)/(2*E^(2*t)), (-2*c1 + c2 - 2*c2*t + c3*t - c3*t^2)/E^(2*t)} ;[o] -2 c2 + c3 - 2 c3 t {-------------------, 2 t 2 E 2 -4 c1 + 4 c2 - 3 c3 - 4 c2 t + 4 c3 t - 2 c3 t -----------------------------------------------, 2 t 2 E 2 -2 c1 + c2 - 2 c2 t + c3 t - c3 t ----------------------------------} 2 t E :[font = input; preserveAspect] solns = Flatten[Table[X[t],{c1,-1,1},{c2,-1,1},{c3,-1,1}],2]; :[font = input; preserveAspect; startGroup] p = 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.79001 1.35872 m .7496 1.2863 L s P p .004 w .64983 .87033 m .62615 .84685 L s P p .004 w .56922 .78229 m .55512 .77687 L s P p .004 w .78558 1.22872 m .76649 1.19734 L s P p .004 w .39105 .31879 m .39828 .36731 L s P p .004 w .57632 1.1294 m .56088 1.10334 L s P p .004 w .53137 .53825 m .52067 .55146 L s P p .004 w .66259 .88379 m .64983 .87033 L s P p .004 w .80109 1.37852 m .79001 1.35872 L s P p .004 w .58726 .79166 m .56922 .78229 L s P p .004 w .39084 .31706 m .39105 .31879 L s P p .004 w .53759 .53322 m .53137 .53825 L s P p .004 w .59684 1.16219 m .57632 1.1294 L s P p .004 w .6055 .80321 m .58726 .79166 L s P p .004 w .62385 1.20375 m .59684 1.16219 L s P p .004 w .64747 1.23914 m .62385 1.20375 L s P p .002 w .4655 .03997 m .89658 .62788 L s .89658 .62788 m .96003 1.51436 L s .96003 1.51436 m .44967 1.00837 L s .44967 1.00837 m .4655 .03997 L s .12587 .18086 m .03997 1.13289 L s .03997 1.13289 m .44967 1.00837 L s .44967 1.00837 m .4655 .03997 L s .4655 .03997 m .12587 .18086 L s P p p .002 w .12587 .18086 m .4655 .03997 L s P p .002 w .17259 .16148 m .17903 .16912 L s P [(-2)] .15971 .1462 .84292 1 Mshowa p .002 w .2923 .11182 m .29853 .11963 L s P [(0)] .27984 .09619 .79718 1 Mshowa p .002 w .41547 .06072 m .42147 .06871 L s P [(2)] .40346 .04475 .75145 1 Mshowa p .001 w .19627 .15166 m .20011 .15626 L s P p .001 w .22007 .14178 m .22389 .14641 L s P p .001 w .24401 .13185 m .2478 .1365 L s P p .001 w .26809 .12186 m .27185 .12653 L s P p .001 w .31665 .10172 m .32036 .10643 L s P p .001 w .34114 .09156 m .34483 .09629 L s P p .001 w .36578 .08134 m .36943 .08609 L s P p .001 w .39055 .07106 m .39418 .07583 L s P p .001 w .14906 .17124 m .15294 .1758 L s P p .001 w .44053 .05033 m .44411 .05514 L s P P p p .002 w .03997 1.13289 m .60628 1.5964 L s P p .002 w .1443 1.21827 m .15389 1.21549 L s P [(-5)] .1251 1.22384 1 -0.28993 Mshowa p .002 w .35191 1.3882 m .36158 1.38566 L s P [(0)] .33259 1.39329 1 -0.26309 Mshowa p .002 w .53235 1.53589 m .54207 1.53355 L s P [(5)] .51292 1.54057 1 -0.2408 Mshowa p .001 w .18828 1.25428 m .19405 1.25264 L s P p .001 w .23099 1.28923 m .23676 1.28762 L s P p .001 w .27246 1.32317 m .27824 1.32159 L s P p .001 w .31275 1.35615 m .31854 1.3546 L s P p .001 w .39 1.41937 m .3958 1.41787 L s P p .001 w .42704 1.44969 m .43285 1.44822 L s P p .001 w .46309 1.4792 m .46891 1.47775 L s P p .001 w .49818 1.50792 m .504 1.50649 L s P p .001 w .09896 1.18117 m .10471 1.17947 L s P p .001 w .05222 1.14291 m .05796 1.14118 L s P p .001 w .56564 1.56314 m .57148 1.56175 L s P p .001 w .59808 1.58969 m .60392 1.58833 L s P P p P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics3D["<<>>"] ;[o] -Graphics3D- :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Problem 3 :[font = text; inactive; preserveAspect; endGroup] For each of the matrices A below, find the fundamental matrix Phi[t] such that Phi[0] is the identity matrix for the system X'[t] == A.X[t] Verify that Phi'[t]==A.Phi[t]. Then use Phi[t] to find a solution satisfying the given initial condition. Compare the result with the solution given by DSolve[]. a) A = {{ 2,-1}, { 1, 2}}; X[0]=={2,-2}; b) A = {{-1, 5,-1}, { 2, 2,-1}, { 2,-4, 5}}; X[0]=={1,-1,2}; ;[s] 19:0,0;25,1;26,0;62,1;68,0;79,1;85,0;124,1;143,0;156,1;173,0;184,1;190,0;296,1;304,0;307,2;309,1;358,2;360,1;435,-1; 3:8,13,9,Times,0,12,0,0,0;9,13,10,Courier,1,12,0,0,0;2,13,9,Times,2,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Solution :[font = text; inactive; preserveAspect] a) ;[s] 1:0,0;3,-1; 1:1,13,9,Times,2,12,0,0,0; :[font = input; preserveAspect] A = {{ 2,-1}, { 1, 2}}; :[font = text; inactive; preserveAspect] We find fundamental solutions as usual: :[font = input; preserveAspect; startGroup] Eigensystem[A] :[font = output; output; inactive; preserveAspect; endGroup] {{2 - I, 2 + I}, {{-I, 1}, {I, 1}}} ;[o] {{2 - I, 2 + I}, {{-I, 1}, {I, 1}}} :[font = input; preserveAspect] Z[t_] = {I,1}E^((2+I)t); :[font = input; preserveAspect; startGroup] X1[t_] = Re[Z[t]]//ComplexExpand :[font = output; output; inactive; preserveAspect; endGroup] {-(E^(2*t)*Sin[t]), E^(2*t)*Cos[t]} ;[o] 2 t 2 t {-(E Sin[t]), E Cos[t]} :[font = input; preserveAspect; startGroup] X2[t_] = Im[Z[t]]//ComplexExpand :[font = output; output; inactive; preserveAspect; endGroup] {E^(2*t)*Cos[t], E^(2*t)*Sin[t]} ;[o] 2 t 2 t {E Cos[t], E Sin[t]} :[font = text; inactive; preserveAspect] A fundamental matrix has columns given by X1[t] and X2[t]. We use Transpose[] to covert these vectors from rows into columns. ;[s] 7:0,0;42,1;47,0;52,1;57,0;66,1;77,0;126,-1; 2:4,13,9,Times,0,12,0,0,0;3,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] Psi[t_] = Transpose[{X1[t], X2[t]}]; Psi[t]//MatrixForm :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{-(E^(2*t)*Sin[t]), E^(2*t)*Cos[t]}, {E^(2*t)*Cos[t], E^(2*t)*Sin[t]}}] ;[o] 2 t 2 t -(E Sin[t]) E Cos[t] 2 t 2 t E Cos[t] E Sin[t] :[font = input; preserveAspect; startGroup] Psi[0]//MatrixForm :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{0, 1}, {1, 0}}] ;[o] 0 1 1 0 :[font = text; inactive; preserveAspect] To get a fundamental matrix Phi[t] such that Phi[0] is the identity matrix, we multiply Psi[t] on the right by the inverse of Psi[0] (in this case we could also just switch the rows of Psi[t]). ;[s] 14:0,0;28,1;34,0;45,1;51,0;88,2;90,1;94,0;126,2;128,1;132,0;185,2;187,1;191,0;194,-1; 3:6,13,9,Times,0,12,0,0,0;5,13,10,Courier,1,12,0,0,0;3,12,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] Phi[t_] = Psi[t].Inverse[Psi[0]]; Phi[t]//MatrixForm :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{E^(2*t)*Cos[t], -(E^(2*t)*Sin[t])}, {E^(2*t)*Sin[t], E^(2*t)*Cos[t]}}] ;[o] 2 t 2 t E Cos[t] -(E Sin[t]) 2 t 2 t E Sin[t] E Cos[t] :[font = input; preserveAspect; startGroup] Phi[0]//MatrixForm :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{1, 0}, {0, 1}}] ;[o] 1 0 0 1 :[font = text; inactive; preserveAspect] Let's check that Phi[t] satisfies the equation: ;[s] 3:0,0;17,1;23,0;48,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] Phi'[t]//Simplify//MatrixForm :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{-(E^(2*t)*(Cos[t] + 2*Sin[t])), E^(2*t)*(2*Cos[t] - Sin[t])}, {E^(2*t)*(2*Cos[t] - Sin[t]), E^(2*t)*(Cos[t] + 2*Sin[t])}}] ;[o] 2 t 2 t -(E (Cos[t] + 2 Sin[t])) E (2 Cos[t] - Sin[t]) 2 t 2 t E (2 Cos[t] - Sin[t]) E (Cos[t] + 2 Sin[t]) :[font = input; preserveAspect; startGroup] A.Phi[t]//Simplify//MatrixForm :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{-(E^(2*t)*(Cos[t] + 2*Sin[t])), E^(2*t)*(2*Cos[t] - Sin[t])}, {E^(2*t)*(2*Cos[t] - Sin[t]), E^(2*t)*(Cos[t] + 2*Sin[t])}}] ;[o] 2 t 2 t -(E (Cos[t] + 2 Sin[t])) E (2 Cos[t] - Sin[t]) 2 t 2 t E (2 Cos[t] - Sin[t]) E (Cos[t] + 2 Sin[t]) :[font = text; inactive; preserveAspect] The solution satisfying the given initial condition is simply: :[font = input; preserveAspect; startGroup] X[t_] = Phi[t].{2,-2}//Simplify :[font = output; output; inactive; preserveAspect; endGroup] {2*E^(2*t)*(Cos[t] + Sin[t]), -2*E^(2*t)*(Cos[t] - Sin[t])} ;[o] 2 t 2 t {2 E (Cos[t] + Sin[t]), -2 E (Cos[t] - Sin[t])} :[font = text; inactive; preserveAspect] Compare with DSolve[]: ;[s] 3:0,0;13,1;21,0;23,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] soln = DSolve[ {x1'[t] == 2x1[t] - 1x2[t], x2'[t] == x1[t] + 2x2[t], x1[0] == 2, x2[0] == -2}, {x1[t],x2[t]},t] :[font = output; output; inactive; preserveAspect; endGroup] {{x1[t] -> -2*(-I/2*E^((2 - I)*t) + I/2*E^((2 + I)*t)) + 2*(E^((2 - I)*t)/2 + E^((2 + I)*t)/2), x2[t] -> 2*(I/2*E^((2 - I)*t) - I/2*E^((2 + I)*t)) - 2*(E^((2 - I)*t)/2 + E^((2 + I)*t)/2)}} ;[o] -I (2 - I) t I (2 + I) t {{x1[t] -> -2 (-- E + - E ) + 2 2 (2 - I) t (2 + I) t E E 2 (---------- + ----------), 2 2 I (2 - I) t I (2 + I) t x2[t] -> 2 (- E - - E ) - 2 2 (2 - I) t (2 + I) t E E 2 (---------- + ----------)}} 2 2 :[font = text; inactive; preserveAspect] Use ComplexExpand[] to make this look like a real-valued function. ;[s] 3:0,0;4,1;19,0;67,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] {x1[t],x2[t]}/.First[soln]//ComplexExpand//Simplify :[font = output; output; inactive; preserveAspect; endGroup] {2*E^(2*t)*(Cos[t] + Sin[t]), -2*E^(2*t)*(Cos[t] - Sin[t])} ;[o] 2 t 2 t {2 E (Cos[t] + Sin[t]), -2 E (Cos[t] - Sin[t])} :[font = text; inactive; preserveAspect] b) ;[s] 1:0,0;3,-1; 1:1,13,9,Times,2,12,0,0,0; :[font = text; inactive; preserveAspect] We follow the same steps as in part a). ;[s] 3:0,0;36,1;38,0;40,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0; :[font = input; preserveAspect] A = {{-1, 5,-1}, { 2, 2,-1}, { 2,-4, 5}}; :[font = text; inactive; preserveAspect] Eigenvalues/vectors: :[font = input; preserveAspect; startGroup] Eigensystem[A] :[font = output; output; inactive; preserveAspect; endGroup] {{-3, 3, 6}, {{-2, 1, 1}, {1, 1, 1}, {-1, -1, 2}}} ;[o] {{-3, 3, 6}, {{-2, 1, 1}, {1, 1, 1}, {-1, -1, 2}}} :[font = text; inactive; preserveAspect] Fundamental solutions: :[font = input; preserveAspect] X1[t_] = {-2, 1, 1}E^(-3t); X2[t_] = { 1, 1, 1}E^( 3t); X3[t_] = {-1,-1, 2}E^( 6t); :[font = text; inactive; preserveAspect] Fundamental matrix: :[font = input; preserveAspect; startGroup] Psi[t_] = Transpose[{X1[t],X2[t],X3[t]}]; Psi[t]//MatrixForm :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{-2/E^(3*t), E^(3*t), -E^(6*t)}, {E^(-3*t), E^(3*t), -E^(6*t)}, {E^(-3*t), E^(3*t), 2*E^(6*t)}}] ;[o] -2 ---- 3 t 3 t 6 t E E -E -3 t 3 t 6 t E E -E -3 t 3 t 6 t E E 2 E :[font = input; preserveAspect; startGroup] Psi[0]//MatrixForm :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{-2, 1, -1}, {1, 1, -1}, {1, 1, 2}}] ;[o] -2 1 -1 1 1 -1 1 1 2 :[font = text; inactive; preserveAspect] Fundamental matrix Phi[t] that is the identity at t==0: ;[s] 5:0,0;19,1;25,0;50,1;54,0;56,-1; 2:3,13,9,Times,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] Phi[t_] = Psi[t].Inverse[Psi[0]]; Phi[t]//MatrixForm :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{2/(3*E^(3*t)) + E^(3*t)/3, -2/(3*E^(3*t)) + E^(3*t)/3 + E^(6*t)/3, E^(3*t)/3 - E^(6*t)/3}, {-1/(3*E^(3*t)) + E^(3*t)/3, 1/(3*E^(3*t)) + E^(3*t)/3 + E^(6*t)/3, E^(3*t)/3 - E^(6*t)/3}, {-1/(3*E^(3*t)) + E^(3*t)/3, 1/(3*E^(3*t)) + E^(3*t)/3 - (2*E^(6*t))/3, E^(3*t)/3 + (2*E^(6*t))/3}}] ;[o] 3 t 3 t 6 t 2 E -2 E E 3 t 6 t ------ + ---- ------ + ---- + ---- E E 3 t 3 3 t 3 3 ---- - ---- 3 E 3 E 3 3 3 t 3 t 6 t -1 E 1 E E 3 t 6 t ------ + ---- ------ + ---- + ---- E E 3 t 3 3 t 3 3 ---- - ---- 3 E 3 E 3 3 3 t 3 t 6 t -1 E 1 E 2 E 3 t 6 t ------ + ---- ------ + ---- - ------ E 2 E 3 t 3 3 t 3 3 ---- + ------ 3 E 3 E 3 3 :[font = input; preserveAspect; startGroup] Phi[0]//MatrixForm :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}] ;[o] 1 0 0 0 1 0 0 0 1 :[font = text; inactive; preserveAspect] Let's check Phi[t] satisfies the equation: ;[s] 3:0,0;12,1;18,0;43,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] Phi'[t]//MatrixForm :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{-2/E^(3*t) + E^(3*t), 2/E^(3*t) + E^(3*t) + 2*E^(6*t), E^(3*t) - 2*E^(6*t)}, {E^(-3*t) + E^(3*t), -E^(-3*t) + E^(3*t) + 2*E^(6*t), E^(3*t) - 2*E^(6*t)}, {E^(-3*t) + E^(3*t), -E^(-3*t) + E^(3*t) - 4*E^(6*t), E^(3*t) + 4*E^(6*t)}}] ;[o] -2 3 t 2 3 t 6 t ---- + E ---- + E + 2 E 3 t 3 t 3 t 6 t E E E - 2 E -3 t 3 t -3 t 3 t 6 t 3 t 6 t E + E -E + E + 2 E E - 2 E -3 t 3 t -3 t 3 t 6 t 3 t 6 t E + E -E + E - 4 E E + 4 E :[font = input; preserveAspect; startGroup] A.Phi[t]//Expand//MatrixForm :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{-2/E^(3*t) + E^(3*t), 2/E^(3*t) + E^(3*t) + 2*E^(6*t), E^(3*t) - 2*E^(6*t)}, {E^(-3*t) + E^(3*t), -E^(-3*t) + E^(3*t) + 2*E^(6*t), E^(3*t) - 2*E^(6*t)}, {E^(-3*t) + E^(3*t), -E^(-3*t) + E^(3*t) - 4*E^(6*t), E^(3*t) + 4*E^(6*t)}}] ;[o] -2 3 t 2 3 t 6 t ---- + E ---- + E + 2 E 3 t 3 t 3 t 6 t E E E - 2 E -3 t 3 t -3 t 3 t 6 t 3 t 6 t E + E -E + E + 2 E E - 2 E -3 t 3 t -3 t 3 t 6 t 3 t 6 t E + E -E + E - 4 E E + 4 E :[font = text; inactive; preserveAspect] The solution satisfying the given initial condition is given by :[font = input; preserveAspect; startGroup] X[t_] = Phi[t].{1,-1,2}//Simplify :[font = output; output; inactive; preserveAspect; endGroup] {4/(3*E^(3*t)) + (2*E^(3*t))/3 - E^(6*t), -2/(3*E^(3*t)) + (2*E^(3*t))/3 - E^(6*t), (2*(-1 + E^(6*t) + 3*E^(9*t)))/(3*E^(3*t))} ;[o] 3 t 3 t 6 t 9 t 4 2 E 6 t -2 2 E 6 t 2 (-1 + E + 3 E ) {------ + ------ - E , ------ + ------ - E , ----------------------} 3 t 3 3 t 3 3 t 3 E 3 E 3 E :[font = text; inactive; preserveAspect] Compare with DSolve[]: ;[s] 3:0,0;13,1;21,0;23,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] DSolve[ {x1'[t] == -x1[t]+5x2[t]-x3[t], x2'[t] == 2x1[t]+2x2[t]-x3[t], x3'[t] == 2x1[t]-4x2[t]+5x3[t], x1[0]==1,x2[0]==-1,x3[0]==2}, {x1[t],x2[t],x3[t]},t]//Simplify :[font = output; output; inactive; preserveAspect; endGroup; endGroup] {{x1[t] -> 4/(3*E^(3*t)) + (2*E^(3*t))/3 - E^(6*t), x2[t] -> -2/(3*E^(3*t)) + (2*E^(3*t))/3 - E^(6*t), x3[t] -> (2*(-1 + E^(6*t) + 3*E^(9*t)))/(3*E^(3*t))}} ;[o] 3 t 3 t 4 2 E 6 t -2 2 E 6 t {{x1[t] -> ------ + ------ - E , x2[t] -> ------ + ------ - E , 3 t 3 3 t 3 3 E 3 E 6 t 9 t 2 (-1 + E + 3 E ) x3[t] -> ----------------------}} 3 t 3 E :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Problem 4 :[font = text; inactive; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450; endGroup] Use the method of variation of parameters to find the general solution of the system X'[t] == A.X[t] + G[t] for each of the matrices A and vector functions G[t] below. Check that your solution is correct by comparing X'[t] and A.X[t]+G[t]. Then find the solution that satisfies the given initial condition and compare the result with the solution given by DSolve[]. a) (see Problem 3a). A = {{ 2,-1}, { 1, 2}}; G[t_] = {t,1}; X[0]=={2,-2}; b) (see Problem 3b). A = {{-1, 5,-1}, { 2, 2,-1}, { 2,-4, 5}}; G[t_] = {1,E^t,E^(-t)}; X[0]=={1,-1,2}; ;[s] 29:0,0;86,1;113,0;138,1;139,0;161,1;165,0;222,1;227,0;232,1;243,0;361,1;369,0;372,3;374,0;381,2;390,3;391,0;392,3;394,1;456,0;457,3;459,0;466,2;475,3;476,0;477,3;478,0;479,1;571,-1; 4:13,13,9,Times,0,12,0,0,0;8,13,10,Courier,1,12,0,0,0;2,13,9,Times,1,12,0,0,0;6,13,9,Times,2,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Solution :[font = text; inactive; preserveAspect; plain; italic] a) :[font = input; preserveAspect] A = {{ 2,-1}, { 1, 2}}; G[t_] = {t,1}; :[font = text; inactive; preserveAspect] A solution to the associated homogeneous problem, X'[t] == A.X[t], was found in Problem 3a : ;[s] 6:0,0;51,1;66,0;81,2;90,3;91,0;94,-1; 4:3,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0;1,13,9,Times,1,12,0,0,0;1,13,9,Times,2,12,0,0,0; :[font = input; preserveAspect; startGroup] X1[t_] = {-E^(2t)Sin[t], E^(2t)Cos[t]}; X2[t_] = { E^(2t)Cos[t], E^(2t)Sin[t]}; Psi[t_] = Transpose[{X1[t],X2[t]}]; IPsi[t_] = Simplify[Inverse[Psi[t]],Trig->True]; IPsi[t]//MatrixForm :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{-(Sin[t]/E^(2*t)), Cos[t]/E^(2*t)}, {Cos[t]/E^(2*t), Sin[t]/E^(2*t)}}] ;[o] Sin[t] Cos[t] -(------) ------ 2 t 2 t E E Cos[t] Sin[t] ------ ------ 2 t 2 t E E :[font = text; inactive; preserveAspect] A particular solution to the non-homogeneous problem is given by Y[t_] = Psi[t].U[t] where U[t] satisfies U'[t] == IPsi[t].G[t]. ;[s] 11:0,0;65,1;73,2;75,1;84,0;91,1;95,0;106,1;116,2;118,1;127,0;129,-1; 3:4,13,9,Times,0,12,0,0,0;5,13,10,Courier,1,12,0,0,0;2,12,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] IPsi[t].G[t] :[font = output; output; inactive; preserveAspect; endGroup] {Cos[t]/E^(2*t) - (t*Sin[t])/E^(2*t), (t*Cos[t])/E^(2*t) + Sin[t]/E^(2*t)} ;[o] Cos[t] t Sin[t] t Cos[t] Sin[t] {------ - --------, -------- + ------} 2 t 2 t 2 t 2 t E E E E :[font = input; preserveAspect; startGroup] U[t_] = Integrate[IPsi[t].G[t], t]//Simplify :[font = output; output; inactive; preserveAspect; endGroup] {(-6*Cos[t] + 5*t*Cos[t] + 8*Sin[t] + 10*t*Sin[t])/(25*E^(2*t)), -(8*Cos[t] + 10*t*Cos[t] + 6*Sin[t] - 5*t*Sin[t])/(25*E^(2*t))} ;[o] -6 Cos[t] + 5 t Cos[t] + 8 Sin[t] + 10 t Sin[t] {-----------------------------------------------, 2 t 25 E -(8 Cos[t] + 10 t Cos[t] + 6 Sin[t] - 5 t Sin[t]) -------------------------------------------------} 2 t 25 E :[font = input; preserveAspect; startGroup] Y[t_] = Psi[t].U[t]//Simplify :[font = output; output; inactive; preserveAspect; endGroup] {(-2*(4 + 5*t))/25, -6/25 + t/5} ;[o] -2 (4 + 5 t) 6 t {------------, -(--) + -} 25 25 5 :[font = text; inactive; preserveAspect] The general solution to the given system is then :[font = input; preserveAspect; startGroup] X[t_] = c1 X1[t] + c2 X2[t] + Y[t] :[font = output; output; inactive; preserveAspect; endGroup] {(-2*(4 + 5*t))/25 + c2*E^(2*t)*Cos[t] - c1*E^(2*t)*Sin[t], -6/25 + t/5 + c1*E^(2*t)*Cos[t] + c2*E^(2*t)*Sin[t]} ;[o] -2 (4 + 5 t) 2 t 2 t {------------ + c2 E Cos[t] - c1 E Sin[t], 25 6 t 2 t 2 t -(--) + - + c1 E Cos[t] + c2 E Sin[t]} 25 5 :[font = text; inactive; preserveAspect] Let's check: :[font = input; preserveAspect; startGroup] X'[t]//Simplify :[font = output; output; inactive; preserveAspect; endGroup] {-2/5 - c1*E^(2*t)*Cos[t] + 2*c2*E^(2*t)*Cos[t] - 2*c1*E^(2*t)*Sin[t] - c2*E^(2*t)*Sin[t], 1/5 + 2*c1*E^(2*t)*Cos[t] + c2*E^(2*t)*Cos[t] - c1*E^(2*t)*Sin[t] + 2*c2*E^(2*t)*Sin[t]} ;[o] 2 2 t 2 t 2 t {-(-) - c1 E Cos[t] + 2 c2 E Cos[t] - 2 c1 E Sin[t] - 5 2 t 1 2 t 2 t c2 E Sin[t], - + 2 c1 E Cos[t] + c2 E Cos[t] - 5 2 t 2 t c1 E Sin[t] + 2 c2 E Sin[t]} :[font = input; preserveAspect; startGroup] A.X[t] + G[t]//Simplify :[font = output; output; inactive; preserveAspect; endGroup] {-2/5 - c1*E^(2*t)*Cos[t] + 2*c2*E^(2*t)*Cos[t] - 2*c1*E^(2*t)*Sin[t] - c2*E^(2*t)*Sin[t], 1/5 + 2*c1*E^(2*t)*Cos[t] + c2*E^(2*t)*Cos[t] - c1*E^(2*t)*Sin[t] + 2*c2*E^(2*t)*Sin[t]} ;[o] 2 2 t 2 t 2 t {-(-) - c1 E Cos[t] + 2 c2 E Cos[t] - 2 c1 E Sin[t] - 5 2 t 1 2 t 2 t c2 E Sin[t], - + 2 c1 E Cos[t] + c2 E Cos[t] - 5 2 t 2 t c1 E Sin[t] + 2 c2 E Sin[t]} :[font = text; inactive; preserveAspect] We now solve for the constants c1 and c2 that satisfy the given initial condition. ;[s] 5:0,0;31,1;33,0;38,1;40,0;83,-1; 2:3,13,9,Times,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] cs = Solve[X[0] == {2,-2}] :[font = output; output; inactive; preserveAspect; endGroup] {{c1 -> -44/25, c2 -> 58/25}} ;[o] 44 58 {{c1 -> -(--), c2 -> --}} 25 25 :[font = text; inactive; preserveAspect] The solution to the initial value problem is thus :[font = input; preserveAspect; startGroup] X[t_] = X[t] /. First[cs] //Simplify :[font = output; output; inactive; preserveAspect; endGroup] {(2*(-4 - 5*t + 29*E^(2*t)*Cos[t] + 22*E^(2*t)*Sin[t]))/25, (-6 + 5*t - 44*E^(2*t)*Cos[t] + 58*E^(2*t)*Sin[t])/25} ;[o] 2 t 2 t 2 (-4 - 5 t + 29 E Cos[t] + 22 E Sin[t]) {----------------------------------------------, 25 2 t 2 t -6 + 5 t - 44 E Cos[t] + 58 E Sin[t] ------------------------------------------} 25 :[font = text; inactive; preserveAspect] The solution given by DSolve[] is: ;[s] 4:0,0;22,1;30,0;34,1;36,-1; 2:2,13,9,Times,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] soln = DSolve[{ x1'[t] == 2x1[t] - x2[t] + t, x2'[t] == x1[t] + 2x2[t] + 1, x1[0] == 2, x2[0] == -2},{x1[t],x2[t]},t] :[font = output; output; inactive; preserveAspect; endGroup] {{x1[t] -> -8/25 + (29/25 + (22*I)/25)*E^((2 - I)*t) + (29/25 - (22*I)/25)*E^((2 + I)*t) - (2*t)/5, x2[t] -> -6/25 + (-22/25 + (29*I)/25)*E^((2 - I)*t) + (-22/25 - (29*I)/25)*E^((2 + I)*t) + t/5}} ;[o] 8 29 22 I (2 - I) t 29 22 I (2 + I) t 2 t {{x1[t] -> -(--) + (-- + ----) E + (-- - ----) E - ---, 25 25 25 25 25 5 6 22 29 I (2 - I) t x2[t] -> -(--) + (-(--) + ----) E + 25 25 25 22 29 I (2 + I) t t (-(--) - ----) E + -}} 25 25 5 :[font = input; preserveAspect; startGroup] {x1[t],x2[t]}/.First[soln]//ComplexExpand//Simplify :[font = output; output; inactive; preserveAspect; endGroup] {(2*(-4 - 5*t + 29*E^(2*t)*Cos[t] + 22*E^(2*t)*Sin[t]))/25, (-6 + 5*t - 44*E^(2*t)*Cos[t] + 58*E^(2*t)*Sin[t])/25} ;[o] 2 t 2 t 2 (-4 - 5 t + 29 E Cos[t] + 22 E Sin[t]) {----------------------------------------------, 25 2 t 2 t -6 + 5 t - 44 E Cos[t] + 58 E Sin[t] ------------------------------------------} 25 :[font = text; inactive; preserveAspect] b) ;[s] 1:0,0;3,-1; 1:1,13,9,Times,2,12,0,0,0; :[font = input; preserveAspect] A = {{-1, 5,-1}, { 2, 2,-1}, { 2,-4, 5}}; G[t_] = {1,E^t,E^(-t)}; :[font = text; inactive; preserveAspect] A solution to the associated homogeneous problem, X'[t] == A.X[t], was found in Problem 3b: ;[s] 6:0,0;51,1;66,0;81,2;90,3;91,0;93,-1; 4:3,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0;1,13,9,Times,1,12,0,0,0;1,13,9,Times,2,12,0,0,0; :[font = input; preserveAspect; startGroup] X1[t_] = {-2, 1, 1}E^(-3t); X2[t_] = { 1, 1, 1}E^( 3t); X3[t_] = {-1,-1, 2}E^( 6t); Psi[t_] = Transpose[{X1[t],X2[t],X3[t]}]; IPsi[t_] = Inverse[Psi[t]]//Simplify; IPsi[t]//MatrixForm :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{-E^(3*t)/3, E^(3*t)/3, 0}, {1/(3*E^(3*t)), 1/(3*E^(3*t)), 1/(3*E^(3*t))}, {0, -1/(3*E^(6*t)), 1/(3*E^(6*t))}}] ;[o] 3 t 3 t -E E ----- ---- 3 3 0 1 1 1 ------ ------ ------ 3 t 3 t 3 t 3 E 3 E 3 E -1 1 ------ ------ 6 t 6 t 0 3 E 3 E :[font = text; inactive; preserveAspect] A particular solution to the non-homogeneous problem is given by Y[t_] = Psi[t].U[t] where U[t] satisfies U'[t] == IPsi[t].G[t]. ;[s] 7:0,0;65,1;84,0;91,1;95,0;106,1;127,0;129,-1; 2:4,13,9,Times,0,12,0,0,0;3,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] IPsi[t].G[t] :[font = output; output; inactive; preserveAspect; endGroup] {-E^(3*t)/3 + E^(4*t)/3, 1/(3*E^(4*t)) + 1/(3*E^(3*t)) + 1/(3*E^(2*t)), 1/(3*E^(7*t)) - 1/(3*E^(5*t))} ;[o] 3 t 4 t -E E 1 1 1 1 1 {----- + ----, ------ + ------ + ------, ------ - ------} 3 3 4 t 3 t 2 t 7 t 5 t 3 E 3 E 3 E 3 E 3 E :[font = input; preserveAspect; startGroup] U[t_] = Integrate[IPsi[t].G[t], t]//Simplify :[font = output; output; inactive; preserveAspect; endGroup] {(E^(3*t)*(-4 + 3*E^t))/36, -(3 + 4*E^t + 6*E^(2*t))/(36*E^(4*t)), -1/(21*E^(7*t)) + 1/(15*E^(5*t))} ;[o] 3 t t t 2 t E (-4 + 3 E ) -(3 + 4 E + 6 E ) -1 1 {----------------, --------------------, ------- + -------} 36 4 t 7 t 5 t 36 E 21 E 15 E :[font = input; preserveAspect; startGroup] Y[t_] = Psi[t].U[t]//Simplify :[font = output; output; inactive; preserveAspect; endGroup] {1/9 - 1/(28*E^t) - (2*E^t)/5, -2/9 - 1/(28*E^t) - (3*E^t)/20, -2/9 - 5/(28*E^t) + E^t/20} ;[o] t t t 1 1 2 E 2 1 3 E 2 5 E {- - ----- - ----, -(-) - ----- - ----, -(-) - ----- + --} 9 t 5 9 t 20 9 t 20 28 E 28 E 28 E :[font = text; inactive; preserveAspect] The general solution to the given system is then :[font = input; preserveAspect; startGroup] X[t_] = c1 X1[t] + c2 X2[t] + c3 X3[t] + Y[t] :[font = output; output; inactive; preserveAspect; endGroup] {1/9 - (2*c1)/E^(3*t) - 1/(28*E^t) - (2*E^t)/5 + c2*E^(3*t) - c3*E^(6*t), -2/9 + c1/E^(3*t) - 1/(28*E^t) - (3*E^t)/20 + c2*E^(3*t) - c3*E^(6*t), -2/9 + c1/E^(3*t) - 5/(28*E^t) + E^t/20 + c2*E^(3*t) + 2*c3*E^(6*t)} ;[o] t 1 2 c1 1 2 E 3 t 6 t {- - ---- - ----- - ---- + c2 E - c3 E , 9 3 t t 5 E 28 E t 2 c1 1 3 E 3 t 6 t -(-) + ---- - ----- - ---- + c2 E - c3 E , 9 3 t t 20 E 28 E t 2 c1 5 E 3 t 6 t -(-) + ---- - ----- + -- + c2 E + 2 c3 E } 9 3 t t 20 E 28 E :[font = text; inactive; preserveAspect] Let's check: :[font = input; preserveAspect; startGroup] X'[t]//Simplify :[font = output; output; inactive; preserveAspect; endGroup] {(6*c1)/E^(3*t) + 1/(28*E^t) - (2*E^t)/5 + 3*c2*E^(3*t) - 6*c3*E^(6*t), (-3*c1)/E^(3*t) + 1/(28*E^t) - (3*E^t)/20 + 3*c2*E^(3*t) - 6*c3*E^(6*t), (-3*c1)/E^(3*t) + 5/(28*E^t) + E^t/20 + 3*c2*E^(3*t) + 12*c3*E^(6*t)} ;[o] t 6 c1 1 2 E 3 t 6 t {---- + ----- - ---- + 3 c2 E - 6 c3 E , 3 t t 5 E 28 E t -3 c1 1 3 E 3 t 6 t ----- + ----- - ---- + 3 c2 E - 6 c3 E , 3 t t 20 E 28 E t -3 c1 5 E 3 t 6 t ----- + ----- + -- + 3 c2 E + 12 c3 E } 3 t t 20 E 28 E :[font = input; preserveAspect; startGroup] A.X[t] + G[t]//Simplify :[font = output; output; inactive; preserveAspect; endGroup] {(6*c1)/E^(3*t) + 1/(28*E^t) - (2*E^t)/5 + 3*c2*E^(3*t) - 6*c3*E^(6*t), (-3*c1)/E^(3*t) + 1/(28*E^t) - (3*E^t)/20 + 3*c2*E^(3*t) - 6*c3*E^(6*t), (-3*c1)/E^(3*t) + 5/(28*E^t) + E^t/20 + 3*c2*E^(3*t) + 12*c3*E^(6*t)} ;[o] t 6 c1 1 2 E 3 t 6 t {---- + ----- - ---- + 3 c2 E - 6 c3 E , 3 t t 5 E 28 E t -3 c1 1 3 E 3 t 6 t ----- + ----- - ---- + 3 c2 E - 6 c3 E , 3 t t 20 E 28 E t -3 c1 5 E 3 t 6 t ----- + ----- + -- + 3 c2 E + 12 c3 E } 3 t t 20 E 28 E :[font = text; inactive; preserveAspect] We now solve for the constants c1, c2, and c3 that satisfy the given initial condition. ;[s] 7:0,0;31,1;33,0;35,1;37,0;43,1;45,0;88,-1; 2:4,13,9,Times,0,12,0,0,0;3,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] cs = Solve[X[0] == {1,-1,2}] :[font = output; output; inactive; preserveAspect; endGroup] {{c1 -> -23/36, c2 -> 37/36, c3 -> 103/105}} ;[o] 23 37 103 {{c1 -> -(--), c2 -> --, c3 -> ---}} 36 36 105 :[font = text; inactive; preserveAspect] The solution to the initial value problem is thus :[font = input; preserveAspect; startGroup] X[t_] = X[t] /. First[cs] //Simplify :[font = output; output; inactive; preserveAspect; endGroup] {1/9 + 23/(18*E^(3*t)) - 1/(28*E^t) - (2*E^t)/5 + (37*E^(3*t))/36 - (103*E^(6*t))/105, -2/9 - 23/(36*E^(3*t)) - 1/(28*E^t) - (3*E^t)/20 + (37*E^(3*t))/36 - (103*E^(6*t))/105, -2/9 - 23/(36*E^(3*t)) - 5/(28*E^t) + E^t/20 + (37*E^(3*t))/36 + (206*E^(6*t))/105} ;[o] t 3 t 6 t 1 23 1 2 E 37 E 103 E {- + ------- - ----- - ---- + ------- - --------, 9 3 t t 5 36 105 18 E 28 E t 3 t 6 t 2 23 1 3 E 37 E 103 E -(-) - ------- - ----- - ---- + ------- - --------, 9 3 t t 20 36 105 36 E 28 E t 3 t 6 t 2 23 5 E 37 E 206 E -(-) - ------- - ----- + -- + ------- + --------} 9 3 t t 20 36 105 36 E 28 E :[font = text; inactive; preserveAspect] The solution given by DSolve[] is: ;[s] 4:0,0;22,1;30,0;34,1;36,-1; 2:2,13,9,Times,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] DSolve[ {x1'[t] == -x1[t] + 5x2[t] - x3[t] + 1, x2'[t] == 2x1[t] + 2x2[t] - x3[t] + E^t, x3'[t] == 2x1[t] - 4x2[t] + 5x3[t] + E^(-t), x1[0]==1, x2[0]==-1, x3[0]==2}, {x1[t],x2[t],x3[t]},t]//Simplify :[font = output; output; inactive; preserveAspect; endGroup; endGroup] {{x1[t] -> 1/9 + 23/(18*E^(3*t)) - 1/(28*E^t) - (2*E^t)/5 + (37*E^(3*t))/36 - (103*E^(6*t))/105, x2[t] -> -2/9 - 23/(36*E^(3*t)) - 1/(28*E^t) - (3*E^t)/20 + (37*E^(3*t))/36 - (103*E^(6*t))/105, x3[t] -> -2/9 - 23/(36*E^(3*t)) - 5/(28*E^t) + E^t/20 + (37*E^(3*t))/36 + (206*E^(6*t))/105}} ;[o] t 3 t 6 t 1 23 1 2 E 37 E 103 E {{x1[t] -> - + ------- - ----- - ---- + ------- - --------, 9 3 t t 5 36 105 18 E 28 E t 3 t 6 t 2 23 1 3 E 37 E 103 E x2[t] -> -(-) - ------- - ----- - ---- + ------- - --------, 9 3 t t 20 36 105 36 E 28 E t 3 t 6 t 2 23 5 E 37 E 206 E x3[t] -> -(-) - ------- - ----- + -- + ------- + --------}} 9 3 t t 20 36 105 36 E 28 E :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Problem 5 :[font = text; inactive; preserveAspect; endGroup] Consider the autonomous system: x'[t] == 1 - x[t]y[t] y'[t] == x[t] - y[t]^3 a) Find all critical points of this system. At each critical point, calculate the corresponding linear system and find the eigenvalues of the coefficient matrix. Then identify the type and stability of the critical point. b) Plot the vector field using PlotVectorField[] with the options ScaleFunction->(Log[50#+1]&) (this makes the arrows bigger near the critical points) and Frame->True. Use the vector field plot in b) to choose a representative sample of several initial points {x0,y0}. Use NDSolve[] to obtain numerical solutions through these points and plot the corresponding trajectories on a single graph. Indicate what happens to the trajectories for increasing t. ;[s] 20:0,0;33,1;87,2;89,0;310,2;312,0;342,1;359,0;376,1;405,0;466,1;477,0;508,2;510,0;571,1;578,0;584,1;593,0;761,1;762,0;764,-1; 3:10,13,9,Times,0,12,0,0,0;7,13,10,Courier,1,12,0,0,0;3,13,9,Times,2,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Solution :[font = input; preserveAspect] F[x_, y_] = 1 - x y; G[x_, y_] = x - y^3; :[font = text; inactive; preserveAspect] a) Find the critical points: ;[s] 2:0,1;2,0;30,-1; 2:1,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0; :[font = input; preserveAspect; startGroup] cp = Solve[{F[x,y] == 0, G[x,y] == 0}] :[font = output; output; inactive; preserveAspect; endGroup] {{x -> -1, y -> -1}, {x -> -I, y -> I}, {x -> I, y -> -I}, {x -> 1, y -> 1}} ;[o] {{x -> -1, y -> -1}, {x -> -I, y -> I}, {x -> I, y -> -I}, {x -> 1, y -> 1}} :[font = text; inactive; preserveAspect] The critical points are thus {-1,-1}, and {1,1}. To analyze them we need to examine the matrix of partial derivative evaluated at each. ;[s] 5:0,0;29,1;36,0;42,1;47,0;136,-1; 2:3,13,9,Times,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] A = {{D[F[x,y],x], D[F[x,y],y]}, {D[G[x,y],x], D[G[x,y],y]}}; A//MatrixForm :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{-y, -x}, {1, -3*y^2}}] ;[o] -y -x 2 1 -3 y :[font = text; inactive; preserveAspect] At the critical point {-1,-1}: ;[s] 3:0,0;22,1;29,0;31,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] A1 = A /. cp[[1]]; A1//MatrixForm :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{1, 1}, {1, -3}}] ;[o] 1 1 1 -3 :[font = input; preserveAspect; startGroup] Eigensystem[A1] :[font = output; output; inactive; preserveAspect; endGroup] {{-1 - 5^(1/2), -1 + 5^(1/2)}, {{2 - 5^(1/2), 1}, {2 + 5^(1/2), 1}}} ;[o] {{-1 - Sqrt[5], -1 + Sqrt[5]}, {{2 - Sqrt[5], 1}, {2 + Sqrt[5], 1}}} :[font = input; preserveAspect; startGroup] N[%] :[font = output; output; inactive; preserveAspect; endGroup] {{-3.23606797749979, 1.23606797749979}, {{-0.2360679774997898, 1.}, {4.23606797749979, 1.}}} ;[o] {{-3.23607, 1.23607}, {{-0.236068, 1.}, {4.23607, 1.}}} :[font = text; inactive; preserveAspect] Since the eigenvalues have opposite signs, the point {-1,-1} is unstable saddle point. At the point {1,1}: ;[s] 5:0,0;53,1;60,0;100,1;105,0;107,-1; 2:3,13,9,Times,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] A2 = A /. cp[[4]]; A2//MatrixForm :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{-1, -1}, {1, -3}}] ;[o] -1 -1 1 -3 :[font = input; preserveAspect; startGroup] Eigensystem[A2] :[font = output; output; inactive; preserveAspect; endGroup] {{-2, -2}, {{1, 1}, {0, 0}}} ;[o] {{-2, -2}, {{1, 1}, {0, 0}}} :[font = text; inactive; preserveAspect] Since there is only one independent eigenvector, the point {1,1} is a node (proper or improper) or a spiral point. It is asymtotically stable since both eigenvalues are negative. ;[s] 3:0,0;59,1;64,0;179,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = text; inactive; preserveAspect] b) ;[s] 1:0,0;3,-1; 1:1,13,9,Times,2,12,0,0,0; :[font = input; preserveAspect; startGroup] ?PlotVectorField :[font = print; inactive; preserveAspect; endGroup] PlotVectorField[f, {x, x0, x1, (xu)}, {y, y0, y1, (yu)}, (options)] produces a vector field plot of the two-dimensional vector function f. :[font = input; preserveAspect; startGroup] v = PlotVectorField[{F[x,y],G[x,y]},{x,-2,2},{y,-2,2}, ScaleFunction->(Log[50#+1]&), Frame->True] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 258] %! %%Creator: Mathematica %%AspectRatio: .91776 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.501983 0.218514 0.45888 0.218514 [ [(-2)] .06495 0 0 2 Msboxa [(-1)] .28347 0 0 2 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Both First[] and Flatten[] are present below only to get rid of extra braces. Evaluate[] keeps Mathematica from complaining about not being able to compile the functions. ;[s] 9:0,0;161,2;168,0;173,2;182,0;234,2;244,0;251,1;262,0;327,-1; 3:5,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0;3,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect] solns1 = Flatten[Table[{x[t],y[t]} /. First[NDSolve[ {x'[t] == 1 - x[t]y[t], y'[t] == x[t] - y[t]^3, x[0] == i, y[0] == j}, {x[t],y[t]},{t,0,10}]], {i,-2,2,4}, {j,-2,2,.5}],1]; :[font = input; preserveAspect] solns2 = Flatten[Table[{x[t],y[t]} /. 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.7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L s P P p p .004 w .8212 .36061 m .84006 .38246 L .84778 .39379 L .85417 .40542 L .85682 .41134 L .8591 .41733 L .86098 .42336 L .86247 .42943 L .86307 .43247 L .86357 .43551 L .86396 .43855 L .86426 .44158 L .86445 .44459 L .86455 .4476 L .86455 .45059 L .86445 .45356 L .86425 .4565 L .86396 .45941 L .86312 .46512 L .86256 .46791 L .86192 .47066 L .86039 .47599 L .85644 .48593 L .8515 .49473 L .84579 .50227 L .83956 .50852 L .833 .51353 L .82631 .51741 L .81963 .5203 L .81309 .52236 L .8099 .52312 L .80677 .52372 L .8037 .52419 L .80071 .52454 L .79925 .52468 L .7978 .52478 L .79637 .52487 L .79497 .52493 L .79358 .52497 L .79222 .525 L .79087 .525 L .78955 .52499 L .78824 .52496 L .78696 .52492 L .7857 .52487 L .78446 .5248 L .7797 .52443 L .77745 .52418 L .77527 .52392 L .77114 .52331 L Mistroke .76376 .52194 L .75201 .51902 L .73721 .51406 L .72943 .51075 L .72538 .50874 L .7233 .50758 L .72224 .50693 L .72171 .50659 L .72145 .50641 L .72132 .50632 L .72126 .50628 L .72123 .50625 L .72122 .50624 L .72121 .50624 L .72121 .50624 L .72121 .50624 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L Mfstroke P P p p .004 w .7212 .60332 m .71422 .55912 L .71105 .53869 L .71021 .53212 L .70969 .52698 L .70953 .52482 L .70947 .52383 L .70942 .52289 L .70938 .52199 L .70935 .52114 L .70934 .52034 L .70933 .51957 L .70933 .51885 L .70934 .51815 L .70936 .5175 L .70938 .51687 L .70941 .51627 L .70945 .5157 L .70954 .51464 L .70966 .51368 L .70979 .5128 L .7101 .51127 L .71046 .51 L .71086 .50895 L .71128 .50807 L .71171 .50735 L .7126 .50627 L .71305 .50587 L .71349 .50555 L .71392 .5053 L .71434 .5051 L .71474 .50495 L .71514 .50484 L .71533 .5048 L .71552 .50476 L .7157 .50474 L .71588 .50472 L .71597 .50471 L .71605 .5047 L .71614 .5047 L .71623 .50469 L .71631 .50469 L .71639 .50469 L .71647 .50469 L .71656 .50469 L .71664 .50469 L .71671 .50469 L .71679 .50469 L .71687 .5047 L .71716 .50472 L Mistroke .71731 .50474 L .71745 .50475 L .71771 .50479 L .71819 .50489 L .71862 .50501 L .71931 .50524 L .72022 .50565 L .72071 .50591 L .72096 .50606 L .72108 .50615 L .72115 .50619 L .72118 .50621 L .72119 .50622 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L Mfstroke P P p p .004 w .62121 .60332 m .62375 .58582 L .62618 .57239 L .63079 .55294 L .63514 .5394 L .63928 .5294 L .64704 .51568 L .6507 .51086 L .65424 .50696 L .65766 .5038 L .66096 .50124 L .66723 .4975 L .6702 .49618 L .67307 .49515 L .67583 .49438 L .67717 .49408 L .67849 .49382 L .67978 .49362 L .68104 .49345 L .68228 .49333 L .68349 .49324 L .68409 .49321 L .68468 .49319 L .68526 .49318 L .68584 .49317 L .68641 .49317 L .68697 .49318 L .68753 .4932 L .68808 .49322 L .68917 .49328 L .69022 .49336 L .69126 .49347 L .69226 .49359 L .69604 .49425 L .69779 .49465 L .69944 .49508 L .70246 .49602 L .70748 .49799 L .71415 .50143 L .71772 .50368 L .71952 .50495 L .72041 .50561 L .72083 .50593 L .72103 .50609 L .72112 .50617 L .72117 .5062 L .72119 .50622 L .7212 .50623 L .7212 .50623 L .7212 .50623 L Mistroke .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L Mfstroke P P p p .004 w .67121 .60332 m .67124 .59452 L .67132 .5869 L .67145 .58021 L .67162 .57429 L .67205 .56425 L .67231 .55995 L .67258 .55605 L .6732 .54921 L .67388 .54341 L .67461 .53843 L .67538 .53412 L .67701 .52704 L .67873 .52149 L .68049 .51708 L .68227 .51352 L .68406 .51063 L .68584 .50829 L .68933 .50482 L .69102 .50357 L .69267 .50257 L .69427 .50177 L .69582 .50116 L .69732 .5007 L .69876 .50037 L .69946 .50025 L .70015 .50015 L .70082 .50008 L .70115 .50005 L .70147 .50002 L .70179 .5 L .70211 .49999 L .70243 .49998 L .70274 .49997 L .70304 .49997 L .70335 .49997 L .70365 .49997 L .70394 .49998 L .70452 .50001 L .70481 .50003 L .70509 .50005 L .70618 .50016 L .70721 .50031 L .70818 .50049 L .70997 .5009 L .71293 .50183 L .71688 .50357 L .71903 .50477 L .72013 .50547 L .72069 .50585 L Mistroke .72096 .50605 L .72109 .50614 L .72115 .50619 L .72118 .50621 L .72119 .50622 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L Mfstroke P P p p .004 w .57121 .60332 m .5807 .57049 L .58884 .5498 L .5961 .53531 L .60272 .52451 L .61458 .50954 L .62514 .49985 L .63474 .49339 L .64357 .48912 L .64774 .48761 L .65176 .48645 L .65563 .48559 L .65751 .48525 L .65935 .48498 L .66116 .48477 L .66206 .48469 L .66294 .48461 L .66381 .48455 L .66468 .48451 L .66554 .48447 L .66639 .48445 L .66723 .48443 L .66806 .48443 L .66889 .48444 L .66971 .48446 L .67051 .48449 L .67131 .48452 L .6721 .48457 L .67289 .48462 L .67594 .48493 L .67741 .48512 L .67885 .48534 L .6843 .48646 L .68922 .48785 L .69365 .48941 L .70107 .4927 L .71101 .49848 L .71629 .50225 L .71891 .5043 L .72015 .50533 L .72072 .50582 L .72098 .50604 L .7211 .50615 L .72116 .50619 L .72118 .50622 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L Mistroke .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L Mfstroke P P p p .004 w .52121 .60332 m .53516 .56859 L .54673 .54662 L .56569 .51947 L .58131 .50303 L .59485 .49212 L .60699 .48462 L .61808 .47949 L .62331 .47762 L .62835 .47614 L .63321 .47501 L .63559 .47456 L .63792 .47419 L .64021 .47388 L .64247 .47365 L .64358 .47355 L .64468 .47347 L .64578 .47341 L .64687 .47336 L .64794 .47333 L .64901 .47331 L .65007 .4733 L .65112 .47331 L .65216 .47333 L .65319 .47336 L .65421 .4734 L .65522 .47346 L .65919 .4738 L .66111 .47404 L .66301 .47431 L .67021 .47575 L .67359 .47665 L .67683 .47764 L .68287 .47986 L .69317 .48476 L .7073 .49395 L .71479 .50013 L .71836 .50343 L .71997 .505 L .72068 .5057 L .72098 .506 L .72111 .50614 L .72116 .50619 L .72119 .50622 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L Mistroke .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L Mfstroke P P p p .004 w .8212 .31206 m .85063 .34024 L .87544 .36676 L .88562 .38014 L .89402 .39379 L .8975 .40074 L .90047 .40777 L .90291 .41486 L .90481 .422 L .90556 .42559 L .90616 .42918 L .90663 .43277 L .90696 .43636 L .90715 .43994 L .9072 .44351 L .90712 .44706 L .9069 .45058 L .90655 .45408 L .90606 .45754 L .90545 .46097 L .90472 .46434 L .9006 .47727 L .8979 .4833 L .89482 .489 L .88769 .49924 L .87959 .50783 L .87086 .51476 L .86182 .52011 L .85271 .52406 L .8482 .52558 L .84375 .52683 L .83936 .52784 L .83507 .52863 L .83087 .52923 L .82677 .52967 L .82476 .52983 L .82279 .52996 L .82084 .53006 L .81892 .53013 L .81703 .53017 L .81516 .53018 L .81333 .53017 L .81153 .53014 L .80976 .53009 L .80802 .53002 L .80463 .52983 L .80135 .52959 L .79819 .52929 L .79222 .52859 L .78157 .52689 L Mistroke .76472 .52314 L .74375 .51667 L .73281 .51232 L .72712 .50965 L .72419 .50809 L .72269 .50722 L .72193 .50674 L .72156 .50649 L .72137 .50636 L .72129 .5063 L .72124 .50626 L .72122 .50625 L .72121 .50624 L .72121 .50624 L .72121 .50624 L .72121 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L Mfstroke P P p p .004 w .47121 .60332 m .48958 .56667 L .50445 .5434 L .52818 .51425 L .54715 .49614 L .56327 .4837 L .57749 .47479 L .5904 .46836 L .60233 .46383 L .608 .46216 L .6135 .46084 L .61885 .45984 L .62405 .45915 L .6266 .45891 L .62786 .45882 L .62912 .45874 L .63036 .45868 L .6316 .45864 L .63283 .45861 L .63405 .4586 L .63526 .4586 L .63647 .45862 L .63766 .45865 L .63885 .4587 L .64003 .45876 L .6412 .45884 L .64353 .45903 L .64582 .45928 L .64807 .45958 L .65249 .46033 L .65677 .46126 L .66091 .46235 L .66878 .46498 L .68267 .4715 L .69395 .47876 L .70264 .48572 L .71337 .49634 L .7182 .50205 L .72013 .5046 L .72085 .50562 L .7211 .50602 L .72118 .50616 L .7212 .50619 L .7212 .50621 L .72121 .50622 L .72121 .50622 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L Mistroke .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .72121 .50623 L .7212 .50623 L .7212 .50623 L Mistroke .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L Mfstroke P P p p .004 w .42121 .55478 m .48147 .49058 L .51916 .46 L .5477 .44079 L .56014 .43375 L .57184 .42805 L .58304 .42358 L .58851 .42178 L .59391 .42025 L .59925 .419 L .60456 .41802 L .6072 .41763 L .60983 .41731 L .61246 .41705 L .61507 .41686 L .61638 .41679 L .61769 .41674 L .61899 .4167 L .6203 .41668 L .6216 .41668 L .6229 .41669 L .6242 .41672 L .6255 .41676 L .6268 .41683 L .62809 .4169 L .63068 .41711 L .63326 .41738 L .63583 .41772 L .64096 .41858 L .64605 .4197 L .65608 .4227 L .66583 .42669 L .67519 .43159 L .68402 .43732 L .69218 .44375 L .706 .45792 L .71598 .47221 L .72216 .48459 L .72404 .48964 L .72528 .49386 L .72569 .49566 L .726 .49727 L .72611 .49801 L .72621 .4987 L .72628 .49935 L .72633 .49996 L .72635 .50025 L .72637 .50053 L .72638 .5008 L .72639 .50106 L Mistroke .72639 .50131 L .72639 .50156 L .72639 .50179 L .72638 .50202 L .72637 .50224 L .72636 .50245 L .72634 .50265 L .72632 .50285 L .72628 .50322 L .72623 .50356 L .7261 .50417 L .72595 .50469 L .72578 .50513 L .7256 .5055 L .72541 .50581 L .72521 .50606 L .72501 .50627 L .72481 .50644 L .72461 .50657 L .72442 .50668 L .72423 .50676 L .72405 .50682 L .72387 .50686 L .72378 .50688 L .7237 .50689 L .72362 .5069 L .72358 .50691 L .72354 .50691 L .7235 .50691 L .72346 .50691 L .72342 .50692 L .72338 .50692 L .72335 .50692 L .72331 .50692 L .72327 .50692 L .72324 .50692 L .72317 .50691 L .72313 .50691 L .7231 .50691 L .72297 .5069 L .72284 .50688 L .72261 .50684 L .72242 .50679 L .72209 .50668 L .72185 .50659 L .72153 .50644 L .72137 .50634 L .72129 .50629 L .72124 .50626 L .72122 .50625 L Mistroke .72121 .50624 L .72121 .50624 L .72121 .50624 L .72121 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L Mfstroke P P p p .004 w .42121 .60332 m .46201 .54014 L .49017 .50886 L .53023 .47455 L .56008 .45548 L .57303 .44908 L .58513 .44425 L .59664 .44078 L .60221 .43951 L .60767 .43854 L .61037 .43817 L .61305 .43786 L .61571 .43762 L .61703 .43753 L .61834 .43745 L .61965 .43739 L .62096 .43735 L .62226 .43733 L .62356 .43732 L .62485 .43732 L .62613 .43735 L .62741 .43739 L .62869 .43744 L .62996 .43751 L .63123 .4376 L .63374 .43782 L .63624 .4381 L .63872 .43845 L .64361 .43931 L .64841 .44041 L .65771 .44325 L .66654 .44688 L .68251 .45603 L .6957 .46665 L .70575 .47726 L .71276 .48652 L .71722 .49369 L .71981 .49872 L .72117 .50198 L .72155 .5031 L .72179 .50396 L .72187 .50431 L .72193 .50462 L .72197 .50489 L .72198 .505 L .722 .50511 L .722 .50517 L .722 .50522 L .72201 .50527 L .72201 .50531 L Mistroke .72201 .50536 L .72201 .5054 L .72201 .50544 L .72201 .50548 L .72201 .50552 L .72201 .50556 L .72201 .50559 L .72201 .50563 L .722 .50566 L .722 .50569 L .72199 .50575 L .72199 .50581 L .72198 .50586 L .72196 .50595 L .72193 .50602 L .7219 .50609 L .72187 .50614 L .72184 .50619 L .72178 .50625 L .72175 .50627 L .72172 .50629 L .72169 .50631 L .72166 .50632 L .72163 .50633 L .72161 .50633 L .72159 .50634 L .72158 .50634 L .72157 .50634 L .72156 .50634 L .72156 .50634 L .72155 .50634 L .72154 .50634 L .72154 .50634 L .72153 .50634 L .72153 .50634 L .72152 .50634 L .72152 .50634 L .72151 .50634 L .7215 .50634 L .72149 .50634 L .72147 .50634 L .72145 .50633 L .72143 .50633 L .7214 .50632 L .72135 .50631 L .72131 .50629 L .72126 .50627 L .72123 .50625 L .72122 .50624 L .72121 .50624 L Mistroke .72121 .50624 L .72121 .50624 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L Mfstroke P P p p .004 w .8212 .26352 m .89424 .33985 L .92117 .37075 L .93138 .38636 L .93557 .39427 L .93909 .40227 L .94194 .41035 L .94409 .41848 L .9449 .42257 L .94553 .42666 L .94598 .43075 L .94625 .43484 L .94634 .43892 L .94626 .44299 L .946 .44703 L .94556 .45105 L .94496 .45503 L .94419 .45896 L .94215 .46667 L .9395 .47412 L .93628 .48123 L .92835 .49426 L .91882 .50539 L .9082 .51448 L .89695 .52154 L .88548 .52675 L .87977 .52875 L .87411 .53039 L .86853 .5317 L .86306 .53273 L .85771 .53351 L .8525 .53408 L .84994 .53428 L .84742 .53445 L .84494 .53457 L .8425 .53465 L .8401 .5347 L .83773 .53472 L .83541 .53471 L .83312 .53467 L .83088 .5346 L .82867 .53451 L .82438 .53428 L .82024 .53397 L .81626 .5336 L .80874 .53273 L .79539 .53065 L .77446 .52615 L .74868 .51854 L .73535 .51345 L Mistroke .72844 .51032 L .72486 .50847 L .72303 .50742 L .7221 .50685 L .72164 .50655 L .72142 .50639 L .72131 .50631 L .72125 .50627 L .72123 .50625 L .72121 .50624 L .72121 .50624 L .72121 .50624 L .72121 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L .7212 .50623 L Mfstroke P P p p .004 w .7212 .21498 m .72893 .23481 L .7365 .25036 L .75133 .274 L .77991 .30696 L .83094 .35554 L .85126 .3787 L .85954 .39063 L .86636 .40288 L .86917 .40911 L .87157 .41541 L .87354 .42176 L .87507 .42815 L .87567 .43135 L .87615 .43456 L .87653 .43775 L .8768 .44095 L .87696 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.50623 L Mfstroke P P P % End of Graphics MathPictureEnd :[font = text; inactive; preserveAspect] It appears that the trajectories near and to the left of the saddle point at {-1,-1} go off to infinity asymptotically along the same curve towards the left. 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m .23838 .04289 L .22877 .06931 L .22041 .08712 L .21287 .09977 L .20587 .10905 L .1992 .11599 L .19272 .12123 L .18631 .12521 L .17986 .12822 L .1733 .13045 L .16995 .13133 L .16655 .13207 L .16308 .13269 L .15954 .13319 L .15592 .13359 L .15408 .13375 L .15221 .1339 L .14841 .13411 L .14647 .13419 L .1445 .13425 L .14251 .13429 L .14049 .13432 L .13843 .13432 L .13635 .13431 L .13424 .13429 L .13209 .13425 L .12769 .13412 L .12316 .13394 L .11847 .1337 L .10862 .13308 L .08674 .1313 L .03204 .12585 L s 0 .12258 m .03204 .12585 L s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w 0 .38627 m .04571 .35456 L .08098 .32634 L .10819 .30027 L .12884 .2758 L .14393 .25284 L .15416 .23161 L .15762 .22176 L .15897 .21705 L .16007 .21248 L .16093 .20807 L .16156 .20381 L .16179 .20174 L .16195 .19972 L .16207 .19773 L .16213 .19578 L .16213 .19387 L .16208 .19201 L .16198 .19018 L .16182 .1884 L .16161 .18665 L .16135 .18495 L .16068 .18166 L .15981 .17853 L .15874 .17556 L .15748 .17273 L .15603 .17005 L .15256 .16511 L .14836 .16067 L .14344 .1567 L .13779 .15312 L .13141 .14989 L .12428 .14695 L .1077 .14176 L .08778 .13717 L .06411 .13288 L .0031 .1243 L s 0 .12394 m .0031 .1243 L s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s P P p p .004 w 1 .46353 m .9867 .47076 L .97308 .47645 L .95943 .48079 L .94598 .48398 L .93939 .4852 L .93291 .4862 L .92656 .487 L .92035 .48763 L .91428 .48811 L .9113 .48829 L .90836 .48844 L .90546 .48856 L .9026 .48866 L .89978 .48872 L .897 .48876 L .89426 .48878 L .89157 .48877 L .88891 .48874 L .8863 .4887 L .88372 .48863 L .88119 .48855 L .87625 .48834 L .87147 .48808 L .86684 .48778 L .85807 .48705 L .8423 .48531 L .8287 .48338 L .79115 .47605 L .77122 .47084 L .7608 .46761 L .75542 .46573 L .75268 .46468 L .75131 .46411 L .75064 .46382 L .7503 .46367 L .75014 .4636 L .75007 .46356 L .75003 .46354 L .75001 .46353 L .75001 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L Mistroke .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L Mfstroke P P p p .004 w 1 .54078 m .86821 .49756 L .80949 .48211 L .78027 .47392 L .76534 .46925 L .7577 .46661 L .75382 .46514 L .75187 .46436 L .75091 .46394 L .75044 .46373 L .75021 .46363 L .7501 .46357 L .75005 .46355 L .75002 .46354 L .75001 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L s P P p p .004 w .875 .61803 m .83622 .55251 L .81274 .52223 L .78573 .49367 L .76302 .47373 L .755 .46739 L .75194 .46504 L .75075 .46412 L .75029 .46376 L .75011 .46362 L .75004 .46356 L .75001 .46354 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L Mistroke .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L Mistroke .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L Mfstroke P P p p .004 w 1 .38627 m .99992 .39002 L .99969 .39374 L .9993 .3974 L .99876 .40102 L .99725 .40808 L .99519 .41489 L .99261 .42142 L .98956 .42764 L .98219 .43907 L .97344 .44906 L .96365 .45757 L .95315 .46463 L .94224 .47034 L .93117 .47483 L .92014 .47826 L .90932 .48078 L .89882 .48255 L .89372 .48319 L .88873 .4837 L .88385 .48408 L .88146 .48423 L .8791 .48435 L .87677 .48445 L .87447 .48453 L .8722 .48458 L .86996 .48461 L .86776 .48463 L .86559 .48463 L .86345 .48461 L .86134 .48457 L .85723 .48446 L .85522 .48438 L .85324 .48429 L .84564 .48383 L .83854 .48324 L .83191 .48256 L .80971 .47939 L .79328 .47621 L .78123 .47341 L .76607 .46924 L .75815 .46668 L .75407 .46521 L .75201 .4644 L .75098 .46397 L .75047 .46375 L .75023 .46363 L .75011 .46358 L .75005 .46355 L .75002 .46354 L Mistroke .75001 .46353 L .75001 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L Mfstroke P P p p .004 w 1 .30902 m 0 0 rlineto 1 .30902 m 1 .30902 L s s s s s s s s s s s s s s s s s s s s 1 .44833 m .99248 .45524 L s .99248 .45524 m .98082 .46347 L .96857 .47015 L .95605 .47543 L .94352 .47946 L .93119 .48244 L .92514 .48358 L .9192 .48452 L .91337 .48529 L .90766 .48589 L .90208 .48634 L .89665 .48667 L .89398 .48679 L .89135 .48688 L .88876 .48695 L .8862 .48699 L .88369 .48702 L .8812 .48702 L .87876 .487 L .87635 .48696 L .87398 .48691 L .87165 .48684 L .86709 .48666 L .86269 .48643 L .85842 .48616 L .84278 .48474 L .82922 .48302 L .81754 .4812 L .79891 .47765 L .77536 .47197 L .76297 .4683 L .75654 .46613 L .75325 .4649 L .75159 .46423 L .75077 .46388 L .75037 .4637 L .75018 .46361 L .75008 .46357 L .75004 .46355 L .75002 .46353 L .75001 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L s P P p p .004 w 1 .61803 m .93952 .55727 L .89985 .52904 L .84912 .50194 L .79732 .48104 L .7734 .47249 L .76163 .46821 L .75575 .46596 L .75282 .46477 L .75137 .46415 L .75066 .46384 L .75031 .46368 L .75015 .4636 L .75007 .46356 L .75003 .46354 L .75002 .46353 L .75001 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L s P P p p .004 w 1 .23176 m 0 0 rlineto 1 .23176 m 1 .23176 L s s s s s s s s s s s s s s s s s s s s s s s s s s s 1 .48483 m .99608 .48591 L s .99608 .48591 m .97973 .48918 L .97174 .49039 L .96391 .49136 L .95625 .49211 L .94878 .49266 L .94511 .49288 L .94149 .49305 L .93793 .49318 L .93441 .49328 L .93095 .49335 L .92753 .49339 L .92417 .4934 L .92086 .49338 L .9176 .49333 L .9144 .49327 L .91125 .49318 L .90814 .49307 L .89625 .49248 L .89061 .49209 L .88517 .49167 L .87485 .4907 L .8564 .48852 L .82702 .48387 L .79001 .47597 L .77056 .47071 L .76044 .46751 L .75523 .46566 L .75259 .46464 L .75126 .46409 L .75061 .46381 L .75029 .46367 L .75014 .46359 L .75007 .46356 L .75003 .46354 L .75001 .46353 L .75001 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L s P P p p .004 w .75 .61803 m .73253 .54768 L .72462 .51519 L .72252 .50473 L .72122 .49655 L .72081 .49311 L .72066 .49153 L .72053 .49003 L .72044 .4886 L .72037 .48726 L .72033 .48597 L .72031 .48476 L .72031 .4836 L .72034 .4825 L .72038 .48145 L .72044 .48045 L .72052 .4795 L .72062 .47859 L .72085 .47691 L .72114 .47537 L .72147 .47398 L .72225 .47154 L .72315 .46952 L .72413 .46785 L .72518 .46645 L .72627 .4653 L .72849 .46358 L .7296 .46294 L .7307 .46244 L .73178 .46203 L .73283 .46172 L .73385 .46148 L .73484 .46131 L .73531 .46124 L .73578 .46119 L .73624 .46114 L .73669 .46111 L .73691 .4611 L .73713 .46109 L .73734 .46108 L .73755 .46107 L .73776 .46107 L .73797 .46106 L .73818 .46106 L .73838 .46106 L .73858 .46106 L .73877 .46107 L .73897 .46107 L .73916 .46108 L .7399 .46111 L Mistroke .74026 .46114 L .7406 .46117 L .74126 .46123 L .74247 .46139 L .74353 .46157 L .74527 .46195 L .74754 .46259 L .74876 .46301 L .74939 .46326 L .7497 .46339 L .74986 .46346 L .74993 .46349 L .74997 .46351 L .74998 .46352 L .74999 .46352 L .75 .46352 L .75 .46352 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L Mfstroke P P p p .004 w .5 .61803 m .50635 .59018 L .51244 .56882 L .52397 .53786 L .53484 .51631 L .54518 .50039 L .56458 .47857 L .57374 .47089 L .58258 .46469 L .59112 .45966 L .59938 .45557 L .61505 .44962 L .62249 .44752 L .62966 .44589 L .63656 .44466 L .63992 .44418 L .64321 .44377 L .64644 .44344 L .6496 .44318 L .65269 .44299 L .65572 .44285 L .65721 .4428 L .65869 .44277 L .66014 .44275 L .66159 .44274 L .66301 .44274 L .66442 .44275 L .66582 .44278 L .6672 .44281 L .6699 .44291 L .67255 .44304 L .67513 .44321 L .67765 .44341 L .6871 .44445 L .69146 .44508 L .69558 .44578 L .70313 .44727 L .71568 .45041 L .73235 .45588 L .74128 .45947 L .7458 .46148 L .74801 .46253 L .74906 .46305 L .74956 .4633 L .7498 .46342 L .74991 .46348 L .74996 .4635 L .74998 .46351 L .74999 .46352 L .75 .46352 L Mistroke .75 .46352 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L Mfstroke P P p p .004 w .625 .61803 m .62507 .60403 L .62528 .5919 L .62561 .58125 L .62603 .57183 L .62711 .55586 L .62775 .54902 L .62845 .5428 L .62999 .53191 L .63169 .52269 L .63352 .51477 L .63544 .50791 L .63952 .49663 L .6438 .48781 L .64821 .48078 L .65267 .47512 L .65714 .47053 L .66159 .46679 L .67031 .46128 L .67454 .45928 L .67866 .45769 L .68267 .45643 L .68655 .45545 L .69029 .45472 L .69389 .4542 L .69564 .454 L .69735 .45384 L .69903 .45372 L .69985 .45368 L .70067 .45364 L .70147 .45361 L .70227 .45358 L .70305 .45357 L .70383 .45356 L .7046 .45355 L .70536 .45356 L .70611 .45356 L .70685 .45358 L .7083 .45362 L .70901 .45365 L .70971 .45369 L .71243 .45386 L .71501 .4541 L .71745 .45438 L .72191 .45503 L .72932 .45652 L .73919 .45929 L .74455 .46119 L .74732 .46231 L .7487 .46291 L Mistroke .74938 .46322 L .74971 .46338 L .74986 .46346 L .74994 .46349 L .74997 .46351 L .74999 .46352 L .74999 .46352 L .75 .46352 L .75 .46352 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L Mfstroke P P p p .004 w .375 .61803 m .39872 .56579 L .41909 .53286 L .43724 .50979 L .45378 .49262 L .48344 .46878 L .50983 .45336 L .53382 .44308 L .55591 .43629 L .56634 .43389 L .57638 .43204 L .58605 .43066 L .59075 .43014 L .59536 .42971 L .59989 .42937 L .60212 .42923 L .60433 .42912 L .60652 .42902 L .60869 .42895 L .61084 .42889 L .61296 .42885 L .61506 .42883 L .61715 .42883 L .61921 .42884 L .62125 .42887 L .62327 .42891 L .62527 .42897 L .62725 .42905 L .62921 .42913 L .63683 .42961 L .64052 .42993 L .64412 .43028 L .65773 .43206 L .67005 .43427 L .68111 .43675 L .69965 .44198 L .72451 .45118 L .73772 .45718 L .74426 .46045 L .74736 .46208 L .74879 .46286 L .74945 .46322 L .74975 .46339 L .74989 .46346 L .74995 .4635 L .74998 .46351 L .74999 .46352 L .75 .46352 L .75 .46352 L .75 .46352 L Mistroke .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L Mfstroke P P p p .004 w .25 .61803 m .28488 .56276 L .3138 .5278 L .36122 .48458 L .40025 .45843 L .43411 .44106 L .46445 .42912 L .49218 .42096 L .50525 .41798 L .51785 .41563 L .53002 .41383 L .53595 .41311 L .54178 .41252 L .54751 .41204 L .55315 .41166 L .55594 .41151 L .5587 .41139 L .56143 .41129 L .56415 .41121 L .56684 .41116 L .56951 .41112 L .57215 .41111 L .57478 .41112 L .57738 .41115 L .57995 .41121 L .58251 .41128 L .58504 .41137 L .59495 .41191 L .59977 .41228 L .6045 .41272 L .62251 .41501 L .63097 .41644 L .63907 .41803 L .65415 .42155 L .67992 .42934 L .71523 .44398 L .73396 .45381 L .7429 .45907 L .74692 .46156 L .74868 .46268 L .74943 .46316 L .74976 .46337 L .7499 .46346 L .74996 .4635 L .74998 .46351 L .74999 .46352 L .75 .46352 L .75 .46352 L .75 .46353 L .75 .46353 L Mistroke .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L Mfstroke P P p p .004 w 1 .15451 m 0 0 rlineto 1 .15451 m 1 .15451 L s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s 1 .50141 m .99909 .50144 L s .99909 .50144 m .99429 .50155 L .98956 .50161 L .98491 .50164 L .98033 .50162 L .97582 .50157 L .97139 .50149 L .96704 .50138 L .95856 .50108 L .95037 .50069 L .94248 .50022 L .92755 .4991 L .90091 .49639 L .8588 .49043 L .80635 .48013 L .77902 .47322 L .7648 .46897 L .75745 .46648 L .75371 .46509 L .75182 .46433 L .75088 .46393 L .75043 .46373 L .7502 .46362 L .7501 .46357 L .75005 .46355 L .75002 .46354 L .75001 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L s P P p p .004 w .125 .61803 m .17091 .55971 L .20811 .52267 L .26743 .47628 L .31486 .44746 L .35514 .42767 L .39071 .41349 L .42298 .40325 L .45281 .39604 L .46699 .39338 L .48074 .39128 L .49411 .38969 L .50711 .38859 L .51348 .38821 L .51664 .38807 L .51977 .38794 L .52288 .38785 L .52598 .38778 L .52905 .38773 L .5321 .38771 L .53513 .38772 L .53815 .38775 L .54114 .3878 L .54411 .38787 L .54706 .38797 L .54999 .38809 L .5558 .3884 L .56152 .3888 L .56716 .38928 L .5782 .39046 L .58891 .39194 L .59927 .39369 L .61892 .39788 L .65365 .40825 L .68187 .41979 L .70359 .43088 L .73042 .44778 L .74249 .45687 L .74732 .46092 L .74911 .46255 L .74974 .46318 L .74994 .46341 L .74998 .46346 L .75 .46349 L .75 .4635 L .75001 .46351 L .75001 .46351 L .75001 .46351 L .75001 .46352 L .75001 .46352 L Mistroke .75001 .46352 L .75001 .46352 L .75001 .46352 L .75001 .46352 L .75001 .46352 L .75001 .46352 L .75001 .46352 L .75001 .46352 L .75001 .46352 L .75001 .46352 L .75001 .46352 L .75001 .46352 L .75001 .46352 L .75001 .46352 L .75001 .46352 L .75001 .46352 L .75001 .46352 L .75001 .46352 L .75001 .46352 L .75001 .46352 L .75001 .46353 L .75001 .46353 L .75001 .46353 L .75001 .46353 L .75001 .46353 L .75001 .46353 L .75001 .46353 L .75001 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L Mistroke .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L Mfstroke P P p p .004 w 0 .54078 m .15065 .43861 L .24488 .38995 L .31622 .35938 L .34733 .34817 L .37659 .3391 L .40458 .33198 L .41825 .32911 L .43174 .32668 L .44511 .32469 L .45837 .32313 L .46497 .32251 L .47155 .322 L .47812 .32159 L .48467 .32129 L .48794 .32118 L .4912 .32109 L .49447 .32103 L .49773 .321 L .50098 .321 L .50423 .32102 L .50748 .32106 L .51073 .32113 L .51397 .32123 L .51721 .32136 L .52368 .32168 L .53013 .32212 L .53657 .32265 L .54938 .32403 L .5621 .32581 L .58718 .33059 L .61156 .33693 L .63496 .34473 L .65703 .35385 L .67743 .36408 L .71198 .38664 L .73693 .40937 L .7524 .42908 L .7571 .43712 L .76018 .44383 L .76123 .4467 L .76199 .44927 L .76228 .45044 L .76251 .45154 L .76269 .45257 L .76282 .45354 L .76287 .454 L .76291 .45445 L .76293 .45488 L .76295 .45529 L Mistroke .76296 .4557 L .76296 .45608 L .76295 .45646 L .76294 .45682 L .76291 .45717 L .76288 .4575 L .76285 .45783 L .7628 .45814 L .76269 .45873 L .76257 .45928 L .76225 .46025 L .76187 .46107 L .76145 .46177 L .76099 .46236 L .76052 .46285 L .76002 .46325 L .75952 .46358 L .75902 .46385 L .75853 .46406 L .75804 .46423 L .75757 .46436 L .75711 .46446 L .75667 .46453 L .75645 .46455 L .75624 .46457 L .75604 .46459 L .75593 .4646 L .75583 .4646 L .75574 .4646 L .75564 .46461 L .75554 .46461 L .75545 .46461 L .75535 .46461 L .75526 .46461 L .75517 .46461 L .75508 .46461 L .7549 .46461 L .75482 .4646 L .75473 .4646 L .7544 .46458 L .75409 .46455 L .75353 .46448 L .75303 .46441 L .75221 .46424 L .7516 .46408 L .75082 .46385 L .75041 .4637 L .7502 .46362 L .7501 .46357 L .75005 .46355 L Mistroke .75002 .46354 L .75001 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L .75 .46353 L Mfstroke P P p p .004 w 0 .61803 m .10199 .51748 L .17239 .4677 L .27256 .4131 L .34719 .38276 L .37954 .37257 L .40982 .36487 L .43857 .35935 L .45249 .35734 L .46617 .3558 L .47292 .3552 L .47961 .35471 L .48625 .35433 L .48955 .35418 L .49284 .35406 L .49612 .35397 L .49938 .3539 L .50263 .35386 L .50587 .35385 L .5091 .35386 L .51231 .35389 L .51552 .35396 L .51871 .35404 L .52188 .35416 L .52505 .3543 L .53134 .35465 L .53759 .3551 L .54378 .35564 L .55601 .35702 L .568 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user PostScript 2. 1.42857 1.81171 1.33575 MAarrowhead1 % End of user PostScript P p .004 w 1 .57389 m .96282 .54514 L s % Start of user PostScript 2. 1.71429 1.85127 1.52824 MAarrowhead1 % End of user PostScript P p .004 w 1 .61803 m .97063 .58173 L s % Start of user PostScript 2. 2. 1.88252 1.76503 MAarrowhead1 % End of user PostScript P P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Problem 6 :[font = text; inactive; preserveAspect] In this problem we consider the equations describing the motion of a pendulum, both damped and undamped. For each situation below convert the equation into a system of equations in x[t] = theta and y[t] = theta'[t]. Plot the correspodining vector field. Use NDSolve[] to solve this system and plot on a single graph the trajectories corresponding to the given initial velocities b. Based on these plots, describe what the pendulum is doing in each case. Experiment to see if you can find a value for the initial velocity b such that the trajectory approaches the unstable equilibrium point {Pi,0} corresponding to the pendulum being stationary in an upright position. ;[s] 13:0,0;181,1;193,0;198,1;214,0;258,1;267,0;379,1;380,0;521,1;522,0;590,1;596,0;668,-1; 2:7,13,9,Times,0,12,0,0,0;6,13,10,Courier,1,12,0,0,0; :[font = text; inactive; preserveAspect; endGroup] a) The undamped pendulum: theta''[t] + Sin[theta] == 0, theta[0] == 0, theta'[0] == b The initial velocities are b = 0.5 to 3.0 in steps of 0.5. b) The damped pendulum: theta''[t] + 0.5 theta'[t] + Sin[theta] == 0, theta[0] == 0, theta'[0] == b The initial velocities are b = 0.5 to 6.0 in steps of 0.5. ;[s] 20:0,2;2,0;26,1;96,0;123,1;130,0;134,1;137,0;150,1;153,0;155,2;158,0;181,1;267,0;294,1;301,0;305,1;308,0;321,1;324,0;326,-1; 3:10,13,9,Times,0,12,0,0,0;8,13,10,Courier,1,12,0,0,0;2,13,9,Times,2,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; 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Graphics["<<>>"] ;[o] -Graphics- :[font = text; inactive; preserveAspect] The first three initial velocities start the pendulum swinging back and forth so that the same x (theta) and y (theta') coordinates are repeated over and over. This translates into the three closed trajectories in the xy-plane. The initial velocities 2, 2.5, and 3.0 are enough to make the pendulum go all the way around, so that the angle (x==theta) is ever increasing. There is an initial velocity which lies on the boundary of these two types of behaviour: namely an initial velocity just enough to swing the pendulum from pointing down (x==0) to pointing up (x==Pi). The trajectory would approach the unstable equilibrium point {Pi,0}. We know that an initial velocity of 2 is too fast. Let's experiment with a smaller values to see if we can `shrink' the trajectory so that it stops at the point {Pi,0}. ;[s] 29:0,0;95,1;96,0;98,1;103,0;109,1;110,0;112,1;118,0;218,1;220,0;251,1;252,0;254,1;257,0;263,1;266,0;341,1;349,0;541,1;545,0;563,1;568,0;632,1;638,0;676,1;677,0;801,1;807,0;809,-1; 2:15,13,9,Times,0,12,0,0,0;14,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] ParametricPlot[Evaluate[{x[t],y[t]} /. 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.09028 .13846 L .09032 .13845 L .09036 .13845 L .09044 .13844 L .09051 .13843 L .09059 .13842 L .09073 .13838 L Mfstroke P P p p .004 w .09091 .30902 m .13555 .28864 L .17405 .26486 L .20582 .2402 L .2309 .21666 L .24977 .19529 L .2631 .17643 L .26789 .1679 L .27155 .15992 L .27414 .15245 L .27506 .14888 L .27543 .14714 L .27574 .14542 L .27599 .14372 L .27618 .14205 L .27631 .14041 L .27639 .13878 L .27641 .13718 L .27638 .1356 L .27629 .13404 L .27615 .13249 L .27596 .13097 L .27571 .12946 L .27506 .1265 L .27422 .1236 L .27317 .12076 L .2705 .11523 L .26708 .1099 L .2581 .09972 L .24639 .09015 L .23213 .08127 L .21555 .07333 L .20647 .06983 L .19694 .06671 L .18701 .06405 L .17675 .06191 L .17152 .06105 L .16624 .06034 L .16358 .06005 L .16091 .05979 L .15823 .05958 L .15555 .0594 L .15286 .05927 L .15016 .05918 L .14747 .05914 L .14477 .05914 L .14208 .05918 L .13938 .05926 L .13669 .05939 L .13401 .05957 L Mistroke .12866 .06005 L .12335 .06071 L .1181 .06154 L .11291 .06255 L 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.07867 .14172 L .07932 .1424 L .08006 .14301 L .08088 .14352 L .08176 .14394 L .08269 .14427 L .08318 .1444 L .08367 .14451 L .08416 .1446 L .08441 .14463 L .08467 .14466 L .08492 .14469 L Mistroke .08517 .1447 L .08543 .14472 L .08568 .14473 L .08594 .14473 L .08619 .14473 L .08644 .14473 L .0867 .14471 L .08695 .1447 L .08721 .14468 L .08771 .14462 L .08821 .14455 L .0887 .14446 L .08967 .14423 L .0906 .14394 L .09232 .1432 L .09382 .14229 L .09505 .14127 L .09555 .14073 L .09598 .14019 L .09634 .13964 L .09662 .1391 L .09673 .13883 L .09682 .13856 L .0969 .1383 L .09695 .13804 L .09698 .13792 L .09699 .13779 L .09701 .13766 L .09702 .13754 L .09702 .13742 L .09702 .1373 L .09702 .13718 L .09701 .13706 L .097 .13695 L .09698 .13683 L .09694 .13661 L .09688 .1364 L .09681 .13619 L .09662 .1358 L .09639 .13545 L .09611 .13514 L .09578 .13487 L .09542 .13463 L .09504 .13444 L .09463 .13428 L .09441 .13422 L .0942 .13417 L .09398 .13413 L .09375 .1341 L .09364 .13408 L Mistroke .09353 .13407 L .09342 .13407 L .0933 .13406 L .09319 .13406 L .09308 .13406 L .09296 .13406 L .09285 .13406 L .09274 .13407 L .09263 .13407 L .0924 .1341 L .09218 .13413 L .09196 .13416 L .09153 .13426 L .09111 .13439 L .09071 .13454 L .08999 .1349 L Mfstroke P P p p .004 w .09091 .37769 m .28975 .25883 L .34539 .22123 L .36688 .20999 L .3766 .20594 L .38584 .20283 L .39031 .20159 L .3947 .20056 L .39903 .19972 L .40331 .19906 L .40543 .1988 L .40755 .19858 L .40966 .1984 L .41176 .19826 L .41386 .19817 L .41595 .19811 L .41805 .19809 L .42014 .1981 L .42224 .19816 L .42434 .19824 L .42854 .19852 L .43066 .19871 L .43278 .19893 L .43704 .19946 L .4457 .20087 L .45458 .20269 L .47322 .20735 L .49329 .21294 L .51495 .21883 L .53819 .22427 L .55034 .22656 L .55653 .22755 L .56278 .22842 L .56909 .22915 L .57544 .22974 L .57863 .22998 L .58182 .23017 L .58503 .23032 L .58823 .23043 L .59145 .2305 L .59466 .23051 L .59787 .23049 L .60108 .23041 L .60429 .23029 L .60749 .23011 L .61068 .22989 L .61387 .22962 L .62649 .22803 L .63271 .22693 L .63884 .22564 L Mistroke .65081 .22247 L .66229 .21857 L .68341 .20887 L .70157 .19729 L .71637 .18461 L .72244 .1781 L .72761 .17158 L .73189 .16511 L .73528 .15877 L .73665 .15565 L .7378 .15259 L .73875 .14957 L .73949 .14661 L .73979 .14515 L .74003 .14371 L .74023 .14228 L .74037 .14087 L .74047 .13948 L .74052 .1381 L .74052 .13674 L .74048 .1354 L .74039 .13408 L .74025 .13277 L .74007 .13149 L .73985 .13022 L .73927 .12775 L .73853 .12535 L .73762 .12304 L .73656 .12082 L .73399 .11663 L .73086 .11281 L .72724 .10937 L .72317 .10633 L .71871 .1037 L .71391 .1015 L .70884 .09972 L .70622 .099 L .70355 .09838 L .70085 .09788 L .69811 .09748 L .69674 .09732 L .69536 .09719 L .69397 .09709 L .69258 .09701 L .69119 .09696 L .6898 .09694 L .6884 .09694 L .68701 .09697 L .68562 .09703 L .68423 .09711 L Mistroke .68147 .09734 L .68009 .0975 L .67872 .09768 L .676 .09811 L .67066 .09924 L .66551 .10072 L .66058 .10251 L .65591 .10458 L .64752 .1094 L .64056 .11487 L .63766 .11776 L .63516 .1207 L .63307 .12366 L .63139 .12661 L .63011 .12951 L .62962 .13094 L .62922 .13235 L .62906 .13304 L .62893 .13373 L .62881 .13441 L .62873 .13509 L .62866 .13575 L .62862 .13641 L .6286 .13707 L .6286 .13771 L .62862 .13834 L .62867 .13897 L .62873 .13959 L .62882 .1402 L .62893 .14079 L .62906 .14138 L .62938 .14253 L .62977 .14363 L .63024 .14469 L .6314 .14667 L .63281 .14846 L .63445 .15004 L .6363 .15142 L .63833 .15258 L .6405 .15353 L .64163 .15392 L .64278 .15426 L .64396 .15455 L .64515 .15479 L .64636 .15497 L .64697 .15504 L .64758 .1551 L .6482 .15515 L .64881 .15518 L .64943 .15521 L Mistroke .65004 .15522 L .65066 .15521 L .65127 .1552 L .65189 .15517 L .6525 .15514 L .65312 .15509 L .65373 .15503 L .65494 .15487 L .65614 .15468 L .65733 .15444 L .65965 .15385 L .66188 .15311 L .66598 .15127 L .66952 .14903 L .67241 .14654 L .67359 .14524 L .67459 .14392 L .67541 .1426 L .67604 .14129 L .67629 .14065 L .6765 .14001 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.6641 .13597 L .6639 .13589 L .6637 .13583 L Mfstroke P P p p .004 w .09091 .41202 m .2242 .34351 L .32001 .28278 L .35723 .26304 L .37405 .25584 L .38999 .25026 L .40526 .24614 L .41271 .24458 L .42006 .24332 L .42733 .24235 L .43454 .24164 L .43813 .24138 L .44172 .24117 L .44529 .24102 L .44887 .24093 L .45244 .24088 L .45601 .24088 L .45958 .24093 L .46315 .24101 L .46673 .24114 L .47031 .2413 L .47749 .24173 L .4847 .24227 L .49196 .2429 L .50661 .24434 L .52147 .24586 L .53654 .24727 L .54414 .24788 L .55178 .2484 L .55561 .24861 L .55945 .24879 L .5633 .24894 L .56715 .24905 L .571 .24912 L .57485 .24915 L .57871 .24914 L .58256 .24908 L .58641 .24898 L .59026 .24882 L .5941 .24861 L .59793 .24835 L .60556 .24768 L .61314 .24677 L .62065 .24564 L .62807 .24428 L .64259 .24087 L .65658 .23656 L .6825 .22557 L .70503 .21215 L .72363 .19732 L Mistroke .73807 .18202 L .7437 .17443 L .7483 .16699 L .75189 .15974 L .75449 .15273 L .75543 .14932 L .75582 .14764 L .75614 .14598 L .75641 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.65961 .14151 L .65976 .1415 L .6599 .14149 L .66004 .14148 L Mistroke .66033 .14145 L .66061 .14141 L .66116 .14129 L .6617 .14114 L .66221 .14096 L .66315 .14052 L .66395 .13998 L .6646 .13939 L .66486 .13909 L .66508 .13878 L .66526 .13847 L .6654 .13817 L .66545 .13802 L .66549 .13787 L .66551 .1378 L .66552 .13772 L .66553 .13765 L .66554 .13758 L .66555 .13751 L .66555 .13744 L .66556 .13737 L .66556 .1373 L .66555 .13724 L .66555 .13717 L .66554 .13711 L .66553 .13704 L .66551 .13692 L .66547 .1368 L .66543 .13668 L .66538 .13657 L .66526 .13636 L .66511 .13618 L .66494 .13601 L .66475 .13587 L .66454 .13575 L .66431 .13565 L .66419 .13561 L .66407 .13558 L .66395 .13555 L .66383 .13553 L .66376 .13552 L .6637 .13551 L .66364 .1355 L .66357 .1355 L .66351 .13549 L .66345 .13549 L .66338 .13549 L .66332 .13549 L .66325 .13549 L .66319 .13549 L Mistroke .66313 .13549 L .66306 .1355 L .66294 .13551 L .66281 .13553 L .66269 .13555 L .66256 .13558 L .66232 .13564 L .6621 .13572 L .66188 .13581 L .66149 .13603 L Mfstroke P P p p .004 w .09091 .44636 m .24098 .37028 L .3002 .33505 L .35084 .30849 L .37375 .29873 L .39546 .29118 L .41626 .28559 L .43642 .2817 L .44632 .2803 L .45614 .27921 L .46589 .27839 L .47075 .27807 L .4756 .2778 L .48044 .27758 L .48527 .2774 L .49009 .27726 L .49492 .27715 L .49973 .27707 L .50455 .277 L .51418 .27692 L .5238 .27686 L .52861 .27683 L .53342 .27679 L .53822 .27673 L .54303 .27665 L .55263 .27643 L .55742 .27627 L .5622 .27608 L .57175 .27556 L .58126 .27486 L .59071 .27395 L .60939 .2714 L .61858 .26973 L .62764 .2678 L .66227 .25735 L .67837 .25062 L .69348 .24305 L .72034 .22611 L .74233 .20791 L .75926 .18965 L .76586 .18076 L .77125 .17214 L .77547 .16381 L .77857 .15582 L .77971 .15196 L .78059 .14818 L .78121 .1445 L .78142 .14269 L .78157 .1409 L .78167 .13913 L Mistroke .7817 .13739 L .78167 .13567 L .78158 .13397 L .78144 .1323 L .78124 .13065 L .78098 .12902 L .78066 .12741 L .77987 .12426 L .77886 .12121 L .77623 .11537 L .77282 .1099 L .76868 .10481 L .76386 .10011 L .75843 .09582 L .75243 .09196 L .74593 .08857 L .739 .08565 L .7317 .08325 L .72794 .08225 L .72411 .08138 L .72022 .08065 L .71629 .08007 L .71432 .07983 L .71233 .07962 L .71034 .07945 L .70834 .07932 L .70634 .07922 L .70433 .07916 L .70233 .07914 L .70032 .07915 L .69831 .07919 L .69631 .07927 L .69231 .07954 L .69032 .07973 L .68834 .07995 L .6844 .08049 L .67666 .08198 L .66916 .08397 L .66196 .08642 L .65513 .08928 L .64278 .09603 L .63244 .10377 L .62431 .11205 L .6185 .12047 L .61646 .12461 L .61565 .12665 L .61498 .12867 L .61446 .13065 L .61406 .13259 L .61391 .13355 L Mistroke .6138 .1345 L .61372 .13544 L .61367 .13637 L .61365 .13728 L .61366 .13819 L .61371 .13908 L .61378 .13996 L .61389 .14082 L .61402 .14167 L .61438 .14334 L .61485 .14495 L .61543 .14649 L .61689 .14941 L .61874 .15206 L .62094 .15444 L .62345 .15653 L .62622 .15834 L .62922 .15985 L .63241 .16106 L .63575 .16197 L .63746 .16231 L .63919 .16258 L .64094 .16278 L .64182 .16285 L .6427 .16291 L .64358 .16295 L .64446 .16297 L .64535 .16297 L .64623 .16295 L .64711 .16292 L .648 .16288 L .64887 .16281 L .64975 .16273 L .6515 .16252 L .65322 .16225 L .65661 .16154 L .65989 .1606 L .66598 .15816 L .66875 .1567 L .67132 .15511 L .67574 .15163 L .67759 .14979 L .67918 .14792 L .6805 .14603 L .68157 .14414 L .68201 .14321 L .68238 .14229 L .68269 .14137 L .68294 .14047 L .68304 .14003 L Mistroke .68312 .13958 L .68319 .13915 L .68325 .13872 L .68329 .13829 L .68331 .13786 L .68332 .13745 L .68332 .13704 L .6833 .13663 L .68327 .13623 L .68317 .13544 L .6831 .13506 L .68301 .13468 L .6828 .13395 L .68254 .13325 L .68224 .13257 L .68149 .13131 L .68058 .13017 L .67952 .12917 L .67833 .1283 L .67703 .12757 L .67565 .12698 L .67492 .12674 L .67418 .12653 L .67343 .12635 L .67267 .12621 L .67228 .12615 L .6719 .1261 L .67151 .12606 L .67112 .12603 L .67073 .126 L .67034 .12598 L .66995 .12597 L .66956 .12597 L .66916 .12598 L .66877 .12599 L .66838 .12601 L .66799 .12604 L .6676 .12607 L .66721 .12612 L .66644 .12622 L .66493 .12651 L .66346 .1269 L .66206 .12738 L .65947 .12858 L .65724 .13001 L .65544 .13161 L .65471 .13244 L .65409 .13328 L .65359 .13412 L .6532 .13495 L Mistroke .65305 .13536 L .65293 .13576 L .65283 .13616 L .65276 .13655 L .65274 .13675 L .65272 .13694 L .65271 .13713 L .65271 .13731 L .65271 .1375 L .65272 .13768 L .65273 .13786 L .65275 .13803 L .65278 .13821 L .65281 .13838 L .6529 .13871 L .653 .13903 L .65313 .13934 L .65344 .13991 L .65383 .14043 L .65429 .14089 L .65481 .14129 L .65538 .14162 L .65599 .1419 L .65631 .14201 L .65663 .14211 L .65696 .1422 L .6573 .14226 L .65764 .14232 L .65799 .14236 L .65816 .14237 L .65833 .14238 L .65851 .14239 L .65868 .14239 L .65886 .14239 L .65903 .14239 L .6592 .14238 L .65938 .14237 L .65955 .14236 L .65972 .14234 L .66007 .1423 L .66041 .14224 L .66074 .14218 L .6614 .14201 L .66203 .1418 L .66263 .14156 L .66371 .14098 L .66462 .14031 L .66501 .13995 L .66534 .13958 L .66563 .13921 L Mistroke .66586 .13883 L .66604 .13846 L .66617 .1381 L .66621 .13792 L .66623 .13783 L .66625 .13775 L .66626 .13766 L .66627 .13758 L .66628 .13749 L .66628 .13741 L .66628 .13733 L .66628 .13724 L .66628 .13716 L .66627 .13708 L .66626 .13701 L .66625 .13693 L .66621 .13678 L .66617 .13664 L .66611 .1365 L .66598 .13624 L .66581 .136 L .66561 .1358 L .66539 .13561 L .66514 .13546 L .66487 .13533 L .66473 .13528 L .66459 .13523 L .66429 .13516 L .66414 .13514 L .66406 .13513 L .66399 .13512 L .66391 .13511 L .66383 .1351 L .66375 .1351 L .66368 .1351 L .6636 .1351 L .66352 .1351 L .66345 .1351 L .66337 .1351 L .66329 .13511 L .66321 .13512 L .66306 .13513 L .66291 .13516 L .66276 .13518 L .66247 .13525 L .66219 .13534 L .66166 .13557 L .66121 .13585 L .66084 .13616 L .66056 .13649 L Mistroke Mfstroke P P p p .004 w .09091 .48069 m .2578 .39734 L .32427 .36101 L .35411 .34691 L .3822 .33566 L .40893 .32705 L .43463 .32075 L .4596 .31632 L .48406 .31327 L .53199 .3092 L .55555 .30718 L .5788 .30456 L .60163 .30101 L .62389 .29629 L .64541 .29028 L .66601 .28299 L .7038 .26513 L .73623 .24436 L .76275 .22261 L .78332 .20141 L .79822 .18165 L .80369 .17244 L .80792 .16369 L .81098 .15539 L .81208 .15141 L .81254 .14945 L .81292 .14753 L .81324 .14562 L .81349 .14375 L .81368 .1419 L .81381 .14007 L .81387 .13827 L .81387 .13649 L .81381 .13473 L .81369 .13299 L .81351 .13128 L .81327 .12959 L .81262 .12627 L .81175 .12304 L .81065 .11989 L .80782 .11381 L .80418 .10803 L .79976 .10256 L .7946 .09738 L .78225 .08797 L .77515 .0838 L .7675 .08002 L .75936 .07667 L .75079 .0738 L .74186 .07146 L Mistroke .73728 .0705 L .73264 .06968 L .72796 .06901 L .7256 .06874 L .72323 .0685 L .72085 .0683 L .71847 .06814 L .71608 .06803 L .71369 .06795 L .71129 .06791 L .7089 .06791 L .70651 .06796 L .70412 .06804 L .70173 .06817 L .69935 .06833 L .69697 .06854 L .6946 .06879 L .68524 .07017 L .68064 .0711 L .67611 .07217 L .65887 .07786 L .65091 .08144 L .64347 .08542 L .63035 .09431 L .61981 .10399 L .61555 .10896 L .61198 .11394 L .6091 .11888 L .60689 .12373 L .60603 .12611 L .60534 .12845 L .6048 .13075 L .60459 .13189 L .60443 .13301 L .6043 .13412 L .6042 .13522 L .60415 .1363 L .60413 .13737 L .60415 .13843 L .60421 .13947 L .6043 .1405 L .60442 .14151 L .60478 .14349 L .60501 .14446 L .60527 .14541 L .60664 .14904 L .6075 .15075 L .60848 .15238 L .61077 .15543 L .61346 .15815 L Mistroke .6165 .16055 L .61984 .16261 L .62345 .16433 L .62727 .1657 L .62925 .16626 L .63126 .16673 L .6333 .16711 L .63536 .1674 L .6364 .16752 L .63744 .16762 L .63848 .16769 L .63953 .16775 L .64058 .16778 L .64163 .16779 L .64268 .16778 L .64373 .16776 L .64478 .16771 L .64582 .16764 L .64791 .16745 L .64997 .16719 L .65202 .16685 L .65603 .16597 L .65991 .16483 L .6636 .16346 L .67037 .16013 L .67612 .15619 L .68073 .15188 L .68258 .14966 L .68413 .14741 L .68536 .14518 L .68587 .14408 L .68629 .14298 L .68664 .1419 L .68692 .14084 L .68703 .14032 L .68713 .13979 L .6872 .13928 L .68726 .13877 L .6873 .13827 L .68732 .13777 L .68733 .13728 L .68732 .13679 L .68729 .13631 L .68725 .13584 L .68719 .13537 L .68711 .13492 L .68691 .13402 L .68666 .13316 L .68634 .13233 L .68597 .13154 L Mistroke .68506 .13006 L .68396 .12872 L .68269 .12755 L .68127 .12653 L .67972 .12568 L .67806 .12499 L .6772 .12471 L .67632 .12447 L .67543 .12427 L .67452 .12411 L .67406 .12404 L .6736 .12398 L .67314 .12394 L .67268 .1239 L .67222 .12387 L .67175 .12385 L .67129 .12384 L .67082 .12384 L .67036 .12385 L .66989 .12387 L .66943 .1239 L .66896 .12393 L .66804 .12403 L .66713 .12416 L .66623 .12432 L .66533 .12451 L .6636 .12499 L .66193 .12556 L .66035 .12624 L .6575 .12782 L .65512 .12964 L .65413 .1306 L .65326 .13159 L .65254 .13259 L .65195 .13358 L .6515 .13457 L .65133 .13505 L .65119 .13553 L .65108 .13601 L .65104 .13624 L .651 .13647 L .65098 .1367 L .65096 .13692 L .65095 .13715 L .65094 .13737 L .65095 .13758 L .65096 .1378 L .65098 .13801 L .65101 .13822 L .65108 .13862 L Mistroke .65118 .13901 L .65131 .13939 L .65147 .13975 L .65185 .14043 L .65232 .14104 L .65286 .14158 L .65348 .14205 L .65416 .14245 L .65489 .14277 L .65527 .1429 L .65565 .14302 L .65605 .14311 L .65645 .14319 L .65686 .14325 L .65706 .14328 L .65727 .1433 L .65747 .14331 L .65768 .14333 L .65788 .14333 L .65809 .14334 L .6583 .14333 L .6585 .14333 L .65871 .14332 L .65892 .14331 L .65933 .14327 L .65973 .14322 L .66014 .14315 L .66053 .14307 L .66131 .14287 L .66206 .14262 L .66343 .142 L .66461 .14125 L .66558 .14041 L .66597 .13997 L .6663 .13953 L .66658 .13909 L .66679 .13865 L .66687 .13843 L .66694 .13822 L .66699 .13801 L .66703 .1378 L .66704 .1377 L .66705 .1376 L .66706 .1375 L .66706 .1374 L .66706 .1373 L .66706 .1372 L .66705 .13711 L .66704 .13701 L .66703 .13692 L Mistroke .66702 .13683 L .66697 .13665 L .66692 .13648 L .66686 .13632 L .66669 .13601 L .66649 .13574 L .66625 .13549 L .66598 .13528 L .66569 .1351 L .66537 .13495 L .6652 .13489 L .66503 .13483 L .66485 .13479 L .66468 .13475 L .6645 .13472 L .66432 .1347 L .66422 .13469 L .66413 .13468 L .66404 .13468 L .66395 .13468 L .66386 .13468 L .66377 .13468 L .66367 .13468 L .66358 .13469 L .66349 .13469 L .6634 .1347 L .66322 .13472 L .66304 .13475 L .66286 .13478 L .66251 .13487 L .66218 .13498 L .66186 .1351 L .66129 .13541 L .6608 .13576 L .66059 .13595 L .66042 .13614 L .66014 .13653 L Mfstroke P P p p .004 w .09091 .51503 m .27466 .4247 L .34854 .38779 L .3821 .37425 L .41403 .36387 L .44471 .3562 L .47448 .3506 L .53215 .34267 L .56021 .33889 L .58771 .33438 L .6145 .32869 L .64041 .32155 L .68876 .30283 L .73128 .27957 L .79589 .2297 L .81818 .20701 L .83456 .18695 L .84578 .1695 L .84967 .16166 L .85122 .15794 L .85251 .15434 L .85357 .15086 L .85438 .14748 L .8547 .14583 L .85497 .14421 L .85518 .14261 L .85533 .14103 L .85543 .13947 L .85548 .13793 L .85547 .13642 L .85541 .13492 L .8553 .13344 L .85515 .13198 L .85468 .12911 L .85437 .1277 L .85401 .1263 L .85315 .12355 L .85088 .11819 L .84788 .11302 L .84418 .10798 L .83473 .09828 L .82268 .08904 L .80814 .08034 L .79131 .07243 L .78211 .0689 L .77244 .06572 L .76237 .06298 L .75196 .06072 L .74664 .0598 L .74127 .05903 L Mistroke .73585 .05841 L .73312 .05816 L .73039 .05796 L .72765 .05779 L .7249 .05767 L .72215 .05759 L .7194 .05755 L .71665 .05756 L .7139 .05761 L .71116 .0577 L .70841 .05784 L .70294 .05826 L .70022 .05854 L .69751 .05886 L .6868 .06058 L .68155 .06171 L .67637 .063 L .66633 .06608 L .64771 .07394 L .63152 .08358 L .61816 .09436 L .60786 .10564 L .60387 .1113 L .60067 .11689 L .59823 .12236 L .59729 .12504 L .59653 .12768 L .59595 .13027 L .59573 .13154 L .59555 .1328 L .59542 .13405 L .59533 .13528 L .59528 .1365 L .59527 .1377 L .5953 .13889 L .59538 .14006 L .59549 .14122 L .59564 .14235 L .59607 .14458 L .59664 .14673 L .59736 .1488 L .59822 .1508 L .60034 .15457 L .60295 .15799 L .60602 .16108 L .60948 .1638 L .6133 .16614 L .61741 .16811 L .62177 .16969 L .62402 .17034 L Mistroke .62631 .17088 L .62864 .17133 L .631 .17169 L .63219 .17183 L .63338 .17194 L .63457 .17204 L .63577 .17211 L .63697 .17215 L .63817 .17218 L .63937 .17218 L .64057 .17216 L .64177 .17211 L .64297 .17204 L .64535 .17184 L .64654 .17171 L .64772 .17156 L .65007 .17119 L .65468 .17021 L .65912 .16894 L .66338 .1674 L .6674 .16562 L .67465 .16147 L .68066 .15675 L .68317 .15425 L .68532 .15171 L .68713 .14915 L .68858 .1466 L .68968 .14409 L .69011 .14285 L .69044 .14163 L .69058 .14103 L .6907 .14044 L .6908 .13984 L .69087 .13926 L .69093 .13868 L .69096 .13811 L .69098 .13755 L .69098 .13699 L .69096 .13644 L .69092 .1359 L .69086 .13536 L .69078 .13484 L .69057 .13381 L .69029 .13282 L .68995 .13186 L .68954 .13095 L .68854 .12923 L .68731 .12769 L .68588 .12633 L .68428 .12514 L Mistroke .68253 .12414 L .68065 .12333 L .67967 .12299 L .67867 .1227 L .67765 .12246 L .67662 .12226 L .67558 .12211 L .67505 .12205 L .67452 .122 L .67399 .12196 L .67346 .12193 L .67293 .12192 L .6724 .12191 L .67187 .12191 L .67133 .12193 L .6708 .12195 L .67027 .12199 L .66922 .12209 L .66817 .12222 L .66713 .1224 L .66611 .12261 L .66411 .12312 L .6622 .12377 L .65866 .12537 L .65562 .12731 L .65315 .12946 L .65214 .13059 L .65129 .13173 L .65059 .13287 L .65005 .134 L .64984 .13455 L .64967 .1351 L .64953 .13565 L .64948 .13591 L .64943 .13618 L .6494 .13644 L .64937 .1367 L .64935 .13696 L .64935 .13721 L .64935 .13746 L .64935 .13771 L .64937 .13795 L .6494 .13819 L .64947 .13866 L .64952 .13889 L .64958 .13911 L .64988 .13997 L .65008 .14037 L .6503 .14076 L .65082 .14147 L Mistroke .65143 .1421 L .65213 .14265 L .65289 .14311 L .65371 .14349 L .65458 .14379 L .65503 .14391 L .65549 .144 L .65595 .14408 L .65618 .14411 L .65642 .14414 L .65665 .14416 L .65689 .14417 L .65712 .14418 L .65736 .14419 L .6576 .14419 L .65783 .14419 L .65807 .14418 L .6583 .14417 L .65877 .14413 L .65901 .14411 L .65924 .14408 L .66016 .14392 L .66061 .14382 L .66105 .1437 L .66191 .14343 L .66273 .1431 L .66421 .14232 L .66547 .14142 L .66599 .14093 L .66646 .14043 L .66717 .13942 L .66742 .13892 L .66752 .13867 L .6676 .13842 L .66767 .13818 L .66772 .13794 L .66774 .13782 L .66775 .13771 L .66776 .13759 L .66777 .13748 L .66777 .13737 L .66777 .13726 L .66777 .13715 L .66776 .13704 L .66775 .13693 L .66773 .13683 L .66769 .13662 L .66763 .13643 L .66756 .13624 L .66738 .13588 L Mistroke .66716 .13556 L .66689 .13527 L .66659 .13502 L .66626 .13481 L .6659 .13463 L .66571 .13456 L .66551 .13449 L .66531 .13444 L .66511 .13439 L .66491 .13436 L .6647 .13433 L .6646 .13432 L .66449 .13431 L .66439 .1343 L .66428 .1343 L .66418 .1343 L .66407 .1343 L .66397 .1343 L .66386 .1343 L .66376 .13431 L .66365 .13432 L .66345 .13434 L .66324 .13437 L .66304 .1344 L .66264 .1345 L .66225 .13462 L .66154 .13492 L .66092 .13529 L .66042 .13571 L Mfstroke P P p p .004 w .09091 .54936 m .29157 .45238 L .37299 .41527 L .41039 .40234 L .44623 .39264 L .5147 .37952 L .57998 .36819 L .61134 .36109 L .64161 .35237 L .69796 .32982 L .74733 .30237 L .78887 .27361 L .84947 .2231 L .88631 .18765 L .9073 .16484 L .91372 .15675 L .91814 .1502 L .91972 .14738 L .92092 .14481 L .92179 .14244 L .9221 .14132 L .92234 .14024 L .92243 .13971 L .9225 .1392 L .92256 .13869 L .92259 .13819 L .92261 .13769 L .92262 .1372 L .92261 .13672 L .92258 .13624 L .92253 .13577 L .92247 .13531 L .9223 .13439 L .92206 .13348 L .92176 .1326 L .92099 .13085 L .91997 .12912 L .91723 .12567 L .91352 .1221 L .9088 .1183 L .89597 .10958 L .87781 .09885 L .85309 .08587 L .82075 .07145 L .8016 .06444 L .78061 .05825 L .76952 .05567 L .75811 .05355 L .7523 .05269 L .74644 .05197 L Mistroke .74053 .05141 L .73757 .05119 L .73459 .05102 L .73161 .05088 L .72863 .05079 L .72565 .05075 L .72266 .05075 L .71968 .05079 L .71669 .05088 L .71372 .05102 L .71074 .05121 L .70482 .05172 L .69894 .05243 L .69312 .05333 L .68736 .05441 L .6761 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Graphics["<<>>"] ;[o] -Graphics- :[font = text; inactive; preserveAspect] Here the critical velocity appears to be around 3.1. For initial velocities below this, the pendulum swings back and forth with ever decreasing angles x (theta) and velocity y (theta') coordinates. For initial velocities above 3.1, the pendulum swings around once, bringing all the angles to a range between Pi and 3Pi. The damping causes the pendulum to slow down and again the velocity decreases to 0 while the the angles approach 2Pi. Let's plot some solutions near the critical velocity. ;[s] 21:0,0;48,1;51,0;151,1;152,0;154,1;159,0;174,1;175,0;177,1;183,0;227,1;230,0;308,1;310,0;315,1;318,0;401,1;402,0;433,1;436,0;492,-1; 2:11,13,9,Times,0,12,0,0,0;10,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] ParametricPlot[Evaluate[{x[t],y[t]} /. 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output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = text; inactive; preserveAspect; endGroup] It looks like an initial velocity of 3.0885098 is close to the critical value. ;[s] 3:0,0;37,1;46,0;79,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Problem 7 :[font = text; inactive; preserveAspect; endGroup] a) Consider the linear system: x'[t] == y[t] + h*x[t], y'[t] == -x[t] + h*y[t] Show that {0,0} is the only critical point of this system for any value of h. For each of the values h==-0.5, 0.0, and 0.5, plot the vector field (using PlotVectorField[] with the options ScaleFunction->(Log[50#+1]&) and Frame->True) and several trajectories (using DSolve[] and ParametricPlot[]). What can you deduce about the stability of the equilibrium point {0,0} corresponding to relatively small perturbations of the system? b) Now consider the system x'[t] == y[t] + h*x[t](x[t]^2 + y[t]^2), y'[t] == -x[t] + h*y[t](x[t]^2 + y[t]^2) Show that {0,0} is the only critical point of this system and that the linear approximation of this system at {0,0} is always a center for any value of h. For each of the values h==-1 and 1 plot the vector field and several trajectories (using NDSolve[] this time); the case h==0 is the same as in a) and is a center. What can you deduce about the stability of the equilibrium point {0,0} corresponding to relatively small perturbations of the system? ;[s] 48:0,2;2,0;31,1;94,0;104,1;109,0;169,1;170,0;195,1;202,0;204,1;207,0;213,1;216,0;247,1;264,0;282,1;310,0;315,1;326,0;360,1;368,0;373,1;389,0;457,1;462,0;526,2;529,0;555,1;648,0;658,1;663,0;758,1;763,0;800,1;801,0;826,1;831,0;836,1;837,0;892,1;901,0;923,1;927,0;946,2;947,0;1031,1;1036,0;1100,-1; 3:24,13,9,Times,0,12,0,0,0;21,13,10,Courier,1,12,0,0,0;3,13,9,Times,2,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Solution :[font = text; inactive; preserveAspect] a) {0,0} is the only critical point since: ;[s] 4:0,0;2,1;4,2;9,1;44,-1; 3:1,13,9,Times,2,12,0,0,0;2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] Solve[{y + h*x == 0, -x + h*y == 0}, {x,y}] :[font = output; output; inactive; preserveAspect; endGroup] {{x -> 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.73794 .38847 L .73801 .38702 L .73806 .38556 L .73809 .38412 L .7381 .38268 L .73809 .38124 L .73806 .37982 L .73802 .37839 L .73796 .37698 L .73788 .37556 L .73767 .37276 L .73739 .36998 L .73663 .36449 L .7356 .35911 L .73282 .34865 L .7291 .33865 L .72452 .32911 L .71308 .31149 L .69907 .2959 L .68307 .28238 L .66559 .27094 L .64713 .26154 L .62814 .25412 L .61857 .25111 L .60903 .24857 L .59955 .24646 L .59017 .24478 L .58093 .24351 L .57638 .24302 L .57187 .24262 L .56742 .24231 L .56521 .24219 L .56302 .2421 L .56084 .24202 L .55976 .24199 L .55868 .24197 L .55761 .24195 L .55654 .24194 L .55547 .24193 L .55441 .24193 L .55335 .24193 L Mistroke .55229 .24193 L .55124 .24195 L .5502 .24196 L .54812 .24201 L .54606 .24208 L .54402 .24217 L .54199 .24228 L .538 .24254 L .53408 .24288 L .53025 .24329 L .52282 .24431 L .51574 .24557 L .50901 .24705 L .49665 .25061 L .48581 .25482 L .4765 .25952 L .46872 .26459 L .46243 .26987 L .45756 .27525 L .45564 .27794 L .45404 .28061 L .45276 .28326 L .45178 .28587 L .4514 .28716 L .45109 .28843 L .45084 .28969 L .45075 .29031 L .45067 .29093 L .4506 .29155 L .45058 .29185 L .45056 .29216 L .45054 .29246 L .45052 .29277 L .45051 .29307 L .45051 .29337 L .4505 .29367 L .45051 .29397 L .45051 .29426 L .45052 .29456 L .45053 .29485 L .45055 .29515 L .45059 .29573 L .45065 .29631 L .45072 .29688 L .4509 .29801 L .45114 .29912 L .45143 .30021 L .45215 .30231 L .45306 .30432 L .45536 .30804 L Mistroke .4582 .31134 L .46148 .31421 L .46508 .31664 L .4689 .31865 L .47284 .32025 L .47482 .3209 L .47681 .32145 L .47878 .32191 L .48074 .32229 L .48267 .32257 L .48362 .32269 L .48456 .32278 L .48549 .32285 L .48596 .32288 L .48642 .32291 L .48687 .32293 L .48732 .32294 L .48755 .32295 L .48777 .32295 L .488 .32296 L .48822 .32296 L .48844 .32296 L .48866 .32296 L .48888 .32296 L .4891 .32296 L .48932 .32296 L .48954 .32296 L .48975 .32295 L .48997 .32295 L .4904 .32293 L .49082 .32291 L .49166 .32287 L .49248 .3228 L .49329 .32273 L .49485 .32253 L .49634 .32228 L .49776 .32198 L .50037 .32127 L .50267 .32041 L .50465 .31944 L .50631 .3184 L .50765 .3173 L .50871 .31619 L Mfstroke P P p p .004 w .70391 .29105 m .68666 .27751 L .66801 .26614 L .64848 .25689 L .62851 .24968 L .6185 .24681 L .60854 .24441 L .59867 .24246 L .58893 .24094 L .58412 .24034 L .57935 .23984 L .57464 .23944 L .56999 .23913 L .56768 .23901 L .56539 .23892 L .56425 .23888 L .56311 .23884 L .56198 .23882 L .56085 .23879 L .55973 .23878 L .55861 .23876 L .5575 .23876 L .55639 .23876 L .55528 .23876 L .55418 .23877 L .55308 .23879 L .55199 .23881 L .54982 .23887 L .54766 .23894 L .54341 .23916 L .53924 .23945 L .53515 .23981 L .52723 .24076 L .51965 .24197 L .50561 .2451 L .49312 .24904 L .48222 .25361 L .47293 .25866 L .46522 .26404 L .45906 .26961 L .45437 .27525 L .45255 .27806 L .45107 .28085 L .44991 .2836 L .44945 .28496 L .44906 .2863 L .44874 .28763 L .4485 .28895 L .4484 .2896 L .44832 .29025 L Mistroke .44826 .29089 L .44823 .29121 L .44821 .29153 L .4482 .29185 L .44818 .29217 L .44817 .29248 L .44817 .2928 L .44817 .29311 L .44817 .29342 L .44818 .29373 L .44819 .29404 L .44821 .29435 L .44822 .29465 L .44827 .29526 L .44834 .29587 L .44842 .29647 L .44862 .29765 L .44887 .29881 L .44954 .30105 L .45041 .3032 L .45146 .30525 L .45267 .30719 L .45334 .30812 L .45369 .30857 L .45386 .30879 L Mfstroke .45422 .30924 m .4544 .30946 L .45478 .30989 L .45554 .31074 L .45716 .31234 L .4607 .31522 L .46455 .31764 L .46859 .31962 L .47273 .32118 L .47481 .3218 L .47689 .32232 L .47895 .32275 L .48098 .32309 L .48198 .32323 L .48298 .32334 L .48396 .32344 L .48494 .32351 L .48542 .32354 L .4859 .32357 L .48638 .32359 L .48661 .3236 L .48685 .3236 L .48709 .32361 L .48732 .32361 L .48755 .32362 L .48779 .32362 L .48802 .32362 L .48825 .32362 L .48848 .32362 L .48871 .32362 L .48894 .32362 L .48916 .32361 L .48962 .3236 L .49006 .32358 L .49051 .32357 L .49138 .32351 L .49224 .32345 L .49391 .32326 L .4955 .32303 L .49702 .32274 L .49846 .3224 L .5011 .3216 L .50341 .32067 L .50539 .31963 L .50704 .31852 L .50836 .31736 L .50937 .31619 L .50977 .31561 L .5101 .31503 L .51035 .31445 L Mistroke .51046 .31417 L .51055 .31389 L .51062 .31361 L .51068 .31333 L .51071 .3132 L .51073 .31306 L .51074 .31293 L .51076 .31279 L .51076 .31273 L .51077 .31266 L .51077 .31259 L .51077 .31253 L .51077 .31246 L .51077 .3124 L .51077 .31233 L .51077 .31227 L .51077 .3122 L .51077 .31214 L .51077 .31207 L .51076 .31201 L .51075 .31188 L .51074 .31176 L .51071 .31151 L .51066 .31127 L .51053 .31079 L .51036 .31034 L .51015 .30991 L Mfstroke P P p p .004 w .67485 .16503 m .66488 .16401 L .65503 .16319 L .65015 .16286 L .64529 .16257 L .64047 .16234 L .63568 .16215 L .6333 .16208 L .63093 .16201 L .62856 .16196 L .62621 .16192 L .62386 .16189 L .62152 .16187 L .61919 .16187 L .61687 .16187 L .61455 .16189 L .61225 .16191 L .60996 .16195 L .60767 .162 L .60314 .16213 L .59864 .1623 L .58976 .16277 L .58104 .16341 L .56413 .16513 L .54795 .16743 L .53251 .17026 L .50401 .17733 L .47878 .18598 L .45691 .19588 L .43839 .20668 L .42318 .21809 L .41117 .22982 L .40631 .23573 L .40219 .24163 L .39879 .24749 L .39607 .25329 L .394 .259 L .39321 .26181 L .39256 .2646 L .39207 .26735 L .39188 .26871 L .39172 .27007 L .3916 .27141 L .39155 .27208 L .39151 .27275 L .39148 .27341 L .39146 .27407 L .39145 .27473 L .39144 .27539 L .39145 .27605 L Mistroke .39146 .2767 L .39148 .27735 L .3915 .27799 L .39158 .27928 L .39169 .28055 L .39184 .28181 L .39201 .28306 L .39244 .28553 L .39299 .28795 L .39366 .29032 L .3953 .29491 L .39734 .29929 L .39974 .30345 L .40549 .31109 L .4123 .31779 L .41992 .32354 L .42811 .32835 L .43666 .33224 L .44536 .33525 L .44971 .33644 L .45404 .33742 L .45832 .33821 L .46255 .33882 L .46463 .33905 L .46669 .33925 L .46872 .3394 L .46973 .33946 L .47074 .33951 L .47173 .33955 L .47223 .33956 L .47272 .33957 L .47321 .33959 L .4737 .33959 L .47419 .3396 L .47468 .33961 L .47516 .33961 L .47565 .33961 L .47613 .3396 L .47661 .3396 L .47708 .33959 L .47756 .33958 L .4785 .33955 L .47944 .33952 L .48037 .33947 L .4822 .33936 L .48399 .33921 L .48575 .33904 L .48915 .33859 L .49241 .33803 L .49842 .33661 L Mistroke .50376 .33485 L .5084 .33282 L .51234 .3306 L .51559 .32824 L .51817 .32581 L .51922 .32458 L .52012 .32335 L .52087 .32213 L .52147 .32092 L .52193 .31973 L .52212 .31914 L .52227 .31856 L .52239 .31798 L .52243 .3177 L .52247 .31741 L .52251 .31713 L .52253 .31685 L .52254 .31671 L .52255 .31657 L .52256 .31644 L .52256 .3163 L .52257 .31616 L .52257 .31602 L .52257 .31589 L .52256 .31575 L .52256 .31562 L .52255 .31548 L .52255 .31535 L .52254 .31522 L .52252 .31495 L .52249 .31469 L .52241 .31417 L .52231 .31366 L .52218 .31316 L .52186 .3122 L .52146 .31128 L .52098 .3104 L .51982 .30879 L .51843 .30737 L .51687 .30615 L .51518 .30512 L .51341 .30429 L .5116 .30364 L Mfstroke P P p p .004 w .64578 .039 m .61457 .04237 L .58471 .04679 L .55626 .05218 L .52926 .05847 L .47979 .07337 L .43651 .09083 L .39947 .11025 L .36863 .13102 L .34382 .15262 L .32479 .17455 L .31734 .18549 L .31121 .19636 L .30634 .20711 L .30436 .21242 L .30268 .21768 L .30129 .22289 L .30018 .22804 L .29973 .2306 L .29934 .23314 L .29903 .23566 L .29878 .23817 L .29867 .23941 L .29859 .24065 L .29852 .24189 L .29847 .24312 L .29843 .24435 L .29841 .24558 L .2984 .24679 L .29841 .24801 L .29843 .24922 L .29847 .25042 L .29852 .25162 L .29859 .25282 L .29877 .25519 L .29901 .25754 L .29967 .26219 L .30054 .26674 L .30291 .27559 L .30607 .28405 L .30996 .29212 L .31967 .30703 L .33154 .32022 L .3451 .33165 L .35991 .34132 L .37555 .34927 L .39163 .35555 L .39972 .35808 L .40781 .36023 L .41583 .36201 L Mistroke .42377 .36343 L .43159 .3645 L .43545 .36491 L .43926 .36525 L .44303 .3655 L .4449 .3656 L .44675 .36568 L .44859 .36575 L .44951 .36577 L .45042 .36579 L .45133 .36581 L .45224 .36582 L .45314 .36582 L .45404 .36583 L .45494 .36582 L .45583 .36582 L .45672 .36581 L .4576 .36579 L .45936 .36575 L .46111 .36569 L .46284 .36561 L .46455 .36552 L .46793 .36529 L .47124 .365 L .47449 .36466 L .48077 .36379 L .48676 .36272 L .49246 .36146 L .50291 .35845 L .51208 .35488 L .51995 .35089 L .52653 .3466 L .53185 .34213 L .53596 .33758 L .53759 .3353 L .53893 .33303 L .54001 .33079 L .54084 .32858 L .54116 .32749 L .54142 .32642 L .54163 .32535 L .54171 .32482 L .54177 .3243 L .54183 .32378 L .54185 .32352 L .54187 .32326 L .54188 .323 L .54189 .32275 L .5419 .32249 L .54191 .32224 L Mistroke .54191 .32198 L .54191 .32173 L .5419 .32148 L .5419 .32123 L .54188 .32098 L .54187 .32073 L .54183 .32024 L .54178 .31975 L .54172 .31926 L .54157 .31831 L .54136 .31737 L .54112 .31645 L .5405 .31467 L .53973 .31297 L .53778 .30982 L .53537 .30703 L .53259 .30461 L .52954 .30255 L .52631 .30085 L .52297 .2995 L .52129 .29895 L .51961 .29848 L .51794 .29809 L .51628 .29778 L .51465 .29754 L .51305 .29736 L Mfstroke P P p p .004 w .55813 .56107 m .60195 .53974 L .63875 .51669 L .66868 .49254 L .692 .46787 L .70129 .4555 L .70907 .44318 L .71538 .43098 L .72029 .41894 L .72224 .41299 L .72387 .4071 L .72517 .40128 L .72616 .39552 L .72654 .39267 L .72685 .38984 L .72698 .38843 L .72709 .38703 L .72718 .38563 L .72725 .38424 L .7273 .38285 L .72734 .38147 L .72736 .38009 L .72736 .37872 L .72735 .37735 L .72731 .37599 L .72726 .37463 L .7272 .37328 L .72701 .37059 L .72676 .36793 L .72607 .36268 L .72513 .35752 L .72394 .35247 L .72089 .34267 L .71697 .33331 L .70684 .31595 L .6941 .30051 L .67931 .28704 L .66296 .27555 L .64555 .26603 L .62752 .25841 L .60927 .25262 L .60018 .25037 L .59116 .24854 L .58226 .2471 L .57786 .24653 L .57351 .24605 L .5692 .24566 L .56494 .24535 L .56283 .24523 L .56073 .24514 L Mistroke .55865 .24506 L .55762 .24503 L .55658 .245 L .55556 .24498 L .55453 .24497 L .55351 .24496 L .55249 .24495 L .55148 .24495 L .55047 .24495 L .54947 .24496 L .54847 .24498 L .54747 .245 L .54648 .24502 L .5445 .24508 L .54255 .24516 L .54061 .24525 L .53679 .2455 L .53304 .24582 L .52577 .24665 L .51882 .24772 L .5122 .24902 L .50592 .25052 L .50292 .25134 L .50145 .25177 L .50072 .25199 L Mfstroke .49928 .25243 m .49858 .25266 L .49717 .25312 L .49444 .25406 L .48923 .25607 L .47993 .26044 L .47209 .26519 L .46568 .27019 L .46065 .27531 L .45864 .27788 L .45694 .28045 L .45555 .28299 L .45445 .28551 L .45401 .28675 L .45363 .28798 L .45333 .2892 L .45309 .2904 L .45299 .291 L .45291 .29159 L .45285 .29218 L .45282 .29247 L .4528 .29276 L .45278 .29305 L .45276 .29334 L .45275 .29363 L .45274 .29392 L .45274 .29421 L .45274 .29449 L .45274 .29478 L .45275 .29506 L .45276 .29534 L .45277 .29562 L .45281 .29618 L .45286 .29674 L .45292 .29728 L .45309 .29837 L .45331 .29943 L .4539 .30149 L .45466 .30346 L .4556 .30534 L .45669 .30712 L .45927 .31038 L .4623 .31324 L .46566 .31568 L .46925 .31771 L .47299 .31935 L .47678 .3206 L .47867 .32109 L .48055 .3215 L .48241 .32182 L Mistroke .48333 .32195 L .48424 .32206 L .48514 .32215 L .48603 .32222 L .48647 .32225 L .48691 .32227 L .48735 .3223 L .48778 .32231 L .488 .32232 L .48821 .32232 L .48842 .32233 L .48864 .32233 L .48885 .32233 L .48906 .32233 L .48927 .32234 L .48948 .32233 L .48969 .32233 L .4899 .32233 L .49011 .32233 L .49031 .32232 L .49072 .32231 L .49113 .32229 L .49193 .32225 L .49272 .32219 L .49425 .32203 L .49571 .32182 L .49711 .32157 L Mfstroke P P p p .004 w .52907 .43504 m .55098 .42438 L .56937 .41285 L .58434 .40078 L .596 .38844 L .60065 .38226 L .60453 .3761 L .60769 .37 L .61015 .36398 L .61112 .361 L .61193 .35806 L .61258 .35515 L .61308 .35227 L .61327 .35085 L .61343 .34943 L .61349 .34873 L .61354 .34802 L .61359 .34732 L .61362 .34663 L .61365 .34593 L .61367 .34524 L .61368 .34455 L .61368 .34387 L .61367 .34318 L .61366 .3425 L .61363 .34182 L .6136 .34115 L .61351 .33981 L .61338 .33847 L .61304 .33585 L .61256 .33327 L .61197 .33074 L .61044 .32584 L .60848 .32116 L .60342 .31248 L .59705 .30476 L .58965 .29803 L .58148 .29228 L .57278 .28752 L .56376 .28371 L .55464 .28082 L .55009 .27969 L .54558 .27878 L .54113 .27806 L .53893 .27777 L .53675 .27753 L .5346 .27734 L .53247 .27719 L .53142 .27713 L .53037 .27708 L Mistroke .52933 .27704 L .52881 .27702 L .52829 .27701 L .52778 .277 L .52727 .27699 L .52676 .27699 L .52625 .27698 L .52574 .27698 L .52524 .27699 L .52473 .27699 L .52423 .277 L .52373 .27701 L .52324 .27702 L .52225 .27705 L .52127 .27709 L .52031 .27714 L .51839 .27726 L .51652 .27742 L .51288 .27783 L .50941 .27837 L .5061 .27902 L .50296 .27977 L .50146 .28018 L .50072 .28039 L .50036 .2805 L Mfstroke .49964 .28072 m .49929 .28084 L .49859 .28107 L .49722 .28154 L .49462 .28254 L .48997 .28473 L .48604 .28711 L .48284 .2896 L .48033 .29216 L .47932 .29345 L .47847 .29473 L .47777 .29601 L .47722 .29726 L .477 .29788 L .47682 .2985 L .47666 .29911 L .47654 .29971 L .4765 .30001 L .47646 .3003 L .47642 .3006 L .47641 .30074 L .4764 .30089 L .47639 .30104 L .47638 .30118 L .47637 .30133 L .47637 .30147 L .47637 .30161 L .47637 .30175 L .47637 .3019 L .47637 .30204 L .47638 .30218 L .47638 .30232 L .4764 .3026 L .47643 .30288 L .47646 .30315 L .47655 .30369 L .47665 .30422 L .47695 .30525 L .47733 .30624 L .4778 .30718 L .47834 .30807 L .47964 .3097 L .48115 .31113 L .48283 .31235 L .48463 .31337 L .48649 .31418 L .48839 .31481 L .48934 .31505 L .49028 .31526 L .49121 .31542 L Mistroke .49166 .31548 L .49212 .31554 L .49257 .31558 L .49302 .31562 L .49324 .31563 L .49346 .31565 L .49367 .31566 L .49389 .31566 L .494 .31567 L .49411 .31567 L .49421 .31567 L .49432 .31567 L .49443 .31568 L .49453 .31568 L .49464 .31568 L .49474 .31568 L .49485 .31567 L .49495 .31567 L .49505 .31567 L .49516 .31567 L .49536 .31566 L .49557 .31566 L .49597 .31563 L .49636 .3156 L .49712 .31553 L .49786 .31542 L .49855 .31529 L Mfstroke P P p p P P p p .004 w .47093 .18299 m .44902 .19366 L .43063 .20518 L .41566 .21725 L .404 .22959 L .39935 .23578 L .39547 .24193 L .39231 .24804 L .38985 .25406 L .38888 .25703 L .38807 .25997 L .38742 .26289 L .38692 .26576 L .38673 .26719 L .38657 .2686 L .38651 .26931 L .38646 .27001 L .38641 .27071 L .38638 .27141 L .38635 .2721 L .38633 .27279 L .38632 .27348 L .38632 .27417 L .38633 .27485 L .38634 .27553 L .38637 .27621 L .3864 .27689 L .38649 .27823 L .38662 .27956 L .38696 .28219 L .38744 .28476 L .38803 .28729 L .38956 .29219 L .39152 .29687 L .39658 .30555 L .40295 .31327 L .41035 .32001 L .41852 .32575 L .42722 .33051 L .43624 .33432 L .44536 .33722 L .44991 .33834 L .45442 .33926 L .45887 .33997 L .46107 .34026 L .46325 .3405 L .4654 .3407 L .46753 .34085 L .46858 .34091 L .46963 .34096 L Mistroke .47067 .341 L .47119 .34101 L .47171 .34102 L .47222 .34103 L .47273 .34104 L .47324 .34105 L .47375 .34105 L .47426 .34105 L .47476 .34105 L .47527 .34104 L .47577 .34104 L .47627 .34103 L .47676 .34102 L .47775 .34099 L .47873 .34095 L .47969 .3409 L .48161 .34077 L .48348 .34062 L .48712 .3402 L .49059 .33967 L .4939 .33902 L .49704 .33827 L .49854 .33785 L .49928 .33764 L .49964 .33753 L Mfstroke .50036 .33731 m .50071 .3372 L .50141 .33697 L .50278 .33649 L .50538 .33549 L .51003 .3333 L .51396 .33093 L .51716 .32843 L .51967 .32587 L .52068 .32458 L .52153 .3233 L .52223 .32203 L .52278 .32077 L .523 .32015 L .52318 .31954 L .52334 .31893 L .52346 .31833 L .5235 .31803 L .52354 .31773 L .52358 .31744 L .52359 .31729 L .5236 .31714 L .52361 .317 L .52362 .31685 L .52363 .31671 L .52363 .31657 L .52363 .31642 L .52363 .31628 L .52363 .31614 L .52363 .316 L .52362 .31585 L .52362 .31571 L .5236 .31543 L .52357 .31516 L .52354 .31488 L .52345 .31434 L .52335 .31381 L .52305 .31278 L .52267 .3118 L .5222 .31086 L .52166 .30997 L .52036 .30833 L .51885 .30691 L .51717 .30569 L .51537 .30467 L .51351 .30385 L .51161 .30322 L .51066 .30298 L .50972 .30278 L .50879 .30262 L Mistroke .50834 .30255 L .50788 .3025 L .50743 .30245 L .50698 .30242 L .50676 .3024 L .50654 .30239 L .50633 .30238 L .50611 .30237 L .506 .30237 L .50589 .30236 L .50579 .30236 L .50568 .30236 L .50557 .30236 L .50547 .30236 L .50536 .30236 L .50526 .30236 L .50515 .30236 L .50505 .30236 L .50495 .30236 L .50484 .30236 L .50464 .30237 L .50443 .30238 L .50403 .3024 L .50364 .30243 L .50288 .30251 L .50214 .30261 L .50145 .30274 L Mfstroke P P p p .004 w .44187 .05697 m .39805 .07829 L .36125 .10134 L .33132 .12549 L .308 .15017 L .29871 .16254 L .29093 .17485 L .28462 .18706 L .27971 .1991 L .27776 .20504 L .27613 .21093 L .27483 .21675 L .27384 .22251 L .27346 .22536 L .27315 .22819 L .27302 .2296 L .27291 .231 L .27282 .2324 L .27275 .2338 L .2727 .23518 L .27266 .23657 L .27264 .23795 L .27264 .23932 L .27265 .24069 L .27269 .24205 L .27274 .2434 L .2728 .24476 L .27299 .24744 L .27324 .2501 L .27393 .25535 L .27487 .26051 L .27606 .26557 L .27911 .27536 L .28303 .28473 L .29316 .30208 L .3059 .31752 L .32069 .331 L .33704 .34248 L .35445 .35201 L .37248 .35962 L .39073 .36542 L .39982 .36766 L .40884 .3695 L .41774 .37093 L .42214 .37151 L .42649 .37199 L .4308 .37238 L .43506 .37268 L .43717 .3728 L .43927 .3729 L Mistroke .44135 .37297 L .44238 .37301 L .44342 .37303 L .44444 .37305 L .44547 .37307 L .44649 .37308 L .44751 .37308 L .44852 .37308 L .44953 .37308 L .45053 .37307 L .45153 .37306 L .45253 .37304 L .45352 .37302 L .4555 .37295 L .45745 .37288 L .45939 .37278 L .46321 .37253 L .46696 .37222 L .47423 .37139 L .48118 .37032 L .4878 .36902 L .49408 .36751 L .49708 .36669 L .49855 .36626 L .49928 .36605 L Mfstroke .50072 .3656 m .50142 .36538 L .50283 .36492 L .50556 .36397 L .51077 .36197 L .52007 .35759 L .52791 .35284 L .53432 .34784 L .53935 .34272 L .54136 .34015 L .54306 .33758 L .54445 .33504 L .54555 .33253 L .54599 .33129 L .54637 .33006 L .54667 .32884 L .54691 .32763 L .54701 .32704 L .54709 .32644 L .54715 .32586 L .54718 .32556 L .5472 .32527 L .54722 .32498 L .54724 .32469 L .54725 .3244 L .54726 .32411 L .54726 .32383 L .54726 .32354 L .54726 .32326 L .54725 .32297 L .54724 .32269 L .54723 .32241 L .54719 .32185 L .54714 .3213 L .54708 .32075 L .54691 .31967 L .54669 .31861 L .5461 .31655 L .54534 .31457 L .5444 .3127 L .54331 .31092 L .54073 .30765 L .5377 .3048 L .53434 .30235 L .53075 .30032 L .52701 .29869 L .52322 .29743 L .52133 .29694 L .51945 .29654 L .51759 .29622 L Mistroke .51667 .29609 L .51576 .29598 L .51486 .29589 L .51397 .29581 L .51353 .29578 L .51309 .29576 L .51265 .29574 L .51222 .29572 L .512 .29572 L .51179 .29571 L .51158 .29571 L .51136 .2957 L .51115 .2957 L .51094 .2957 L .51073 .2957 L .51052 .2957 L .51031 .2957 L .5101 .2957 L .50989 .29571 L .50969 .29571 L .50928 .29572 L .50887 .29574 L .50807 .29578 L .50728 .29584 L .50575 .296 L .50429 .29621 L .50289 .29647 L Mfstroke P P p p .004 w .35422 .57903 m .38543 .57567 L .41529 .57124 L .44374 .56585 L .47074 .55957 L .52021 .54467 L .56349 .5272 L .60053 .50779 L .63137 .48701 L .65618 .46541 L .67521 .44349 L .68266 .43254 L .68879 .42167 L .69366 .41093 L .69564 .40562 L .69732 .40035 L .69871 .39514 L .69982 .38999 L .70027 .38743 L .70066 .3849 L .70097 .38237 L .70122 .37987 L .70133 .37862 L .70141 .37738 L .70148 .37614 L .70153 .37491 L .70157 .37368 L .70159 .37246 L .7016 .37124 L .70159 .37003 L .70157 .36882 L .70153 .36761 L .70148 .36641 L .70141 .36522 L .70123 .36284 L .70099 .36049 L .70033 .35585 L .69946 .35129 L .69709 .34245 L .69393 .33398 L .69004 .32591 L .68033 .31101 L .66846 .29782 L .6549 .28638 L .64009 .27671 L .62445 .26876 L .60837 .26249 L .60028 .25995 L .59219 .2578 L .58417 .25603 L Mistroke .57623 .25461 L .56841 .25353 L .56455 .25312 L .56074 .25279 L .55697 .25253 L .5551 .25243 L .55325 .25235 L .55141 .25229 L .55049 .25226 L .54958 .25224 L .54867 .25223 L .54776 .25222 L .54686 .25221 L .54596 .25221 L .54506 .25221 L .54417 .25222 L .54328 .25223 L .5424 .25224 L .54064 .25229 L .53889 .25234 L .53716 .25242 L .53545 .25251 L .53207 .25274 L .52876 .25303 L .52551 .25338 L .51923 .25424 L .51324 .25531 L .50754 .25657 L .49709 .25959 L .48792 .26315 L .48005 .26714 L .47347 .27143 L .46815 .2759 L .46404 .28046 L .46241 .28274 L .46107 .285 L .45999 .28724 L .45916 .28945 L .45884 .29054 L .45858 .29162 L .45837 .29268 L .45829 .29321 L .45823 .29374 L .45817 .29426 L .45815 .29452 L .45813 .29477 L .45812 .29503 L .45811 .29529 L .4581 .29554 L .45809 .2958 L Mistroke .45809 .29605 L .45809 .2963 L .4581 .29655 L .4581 .29681 L .45812 .29705 L .45813 .2973 L .45817 .2978 L .45822 .29829 L .45828 .29877 L .45843 .29973 L .45864 .30067 L .45888 .30159 L .4595 .30337 L .46027 .30507 L .46222 .30821 L .46463 .311 L .46741 .31343 L .47046 .31549 L .47369 .31719 L .47703 .31854 L .47871 .31908 L .48039 .31955 L .48206 .31994 L .48372 .32026 L .48535 .3205 L .48695 .32067 L Mfstroke P P p p .004 w .32515 .45301 m .33512 .45403 L .34497 .45485 L .34985 .45518 L .35471 .45546 L .35953 .4557 L .36432 .45588 L .3667 .45596 L .36907 .45602 L .37144 .45607 L .37379 .45611 L .37614 .45614 L .37848 .45616 L .38081 .45617 L .38313 .45616 L .38545 .45615 L .38775 .45612 L .39004 .45608 L .39233 .45603 L .39686 .4559 L .40136 .45573 L .41024 .45526 L .41896 .45463 L .43587 .45291 L .45205 .45061 L .46749 .44778 L .49599 .4407 L .52122 .43205 L .54309 .42216 L .56161 .41136 L .57682 .39995 L .58883 .38821 L .59369 .3823 L .59781 .3764 L .60121 .37054 L .60393 .36474 L .606 .35904 L .60679 .35622 L .60744 .35344 L .60793 .35068 L .60812 .34932 L .60828 .34797 L .6084 .34662 L .60845 .34595 L .60849 .34529 L .60852 .34462 L .60854 .34396 L .60855 .3433 L .60856 .34264 L .60855 .34199 L Mistroke .60854 .34134 L .60852 .34069 L .6085 .34004 L .60842 .33876 L .60831 .33748 L .60816 .33622 L .60799 .33497 L .60756 .3325 L .60701 .33008 L .60634 .32771 L .6047 .32312 L .60266 .31874 L .60026 .31458 L .59451 .30695 L .5877 .30025 L .58008 .2945 L .57189 .28969 L .56334 .28579 L .55464 .28279 L .55029 .2816 L .54596 .28061 L .54168 .27982 L .53745 .27922 L .53537 .27898 L .53331 .27879 L .53128 .27864 L .53027 .27858 L .52926 .27853 L .52827 .27849 L .52777 .27847 L .52728 .27846 L .52679 .27845 L .5263 .27844 L .52581 .27843 L .52532 .27843 L .52484 .27843 L .52435 .27843 L .52387 .27843 L .52339 .27844 L .52292 .27844 L .52244 .27845 L .5215 .27848 L .52056 .27852 L .51963 .27856 L .5178 .27867 L .51601 .27882 L .51425 .279 L .51085 .27945 L .50759 .28001 L .50158 .28143 L Mistroke .49624 .28319 L .4916 .28521 L .48766 .28744 L .48441 .28979 L .48183 .29223 L .48078 .29346 L .47988 .29468 L .47913 .29591 L .47853 .29711 L .47807 .29831 L .47788 .29889 L .47773 .29948 L .47761 .30005 L .47757 .30034 L .47753 .30062 L .47749 .3009 L .47747 .30118 L .47746 .30132 L .47745 .30146 L .47744 .3016 L .47744 .30174 L .47743 .30187 L .47743 .30201 L .47743 .30215 L .47744 .30228 L .47744 .30242 L .47745 .30255 L .47745 .30268 L .47746 .30282 L .47748 .30308 L .47751 .30335 L .47759 .30386 L .47769 .30437 L .47782 .30487 L .47814 .30584 L .47854 .30676 L .47902 .30763 L .48018 .30925 L .48157 .31067 L .48313 .31189 L .48482 .31291 L .48659 .31375 L .4884 .3144 L Mfstroke P P p p .004 w .29609 .32698 m .31334 .34052 L .33199 .35189 L .35152 .36114 L .37149 .36835 L .3815 .37122 L .39146 .37362 L .40133 .37557 L .41107 .37709 L .41588 .37769 L .42065 .3782 L .42536 .3786 L .43001 .3789 L .43232 .37902 L .43461 .37912 L .43575 .37916 L .43689 .37919 L .43802 .37922 L .43915 .37924 L .44027 .37926 L .44139 .37927 L .4425 .37927 L .44361 .37928 L .44472 .37927 L .44582 .37926 L .44692 .37924 L .44801 .37922 L .45018 .37917 L .45234 .37909 L .45659 .37888 L .46076 .37859 L .46485 .37822 L .47277 .37727 L .48035 .37606 L .49439 .37293 L .50688 .369 L .51778 .36443 L .52707 .35938 L .53478 .354 L .54094 .34842 L .54563 .34278 L .54745 .33997 L .54893 .33719 L .55009 .33444 L .55055 .33308 L .55094 .33173 L .55126 .3304 L .5515 .32908 L .5516 .32843 L .55168 .32778 L Mistroke .55174 .32714 L .55177 .32682 L .55179 .3265 L .5518 .32618 L .55182 .32587 L .55183 .32555 L .55183 .32524 L .55183 .32493 L .55183 .32461 L .55182 .3243 L .55181 .32399 L .55179 .32369 L .55178 .32338 L .55173 .32277 L .55166 .32217 L .55158 .32157 L .55138 .32038 L .55113 .31923 L .55046 .31698 L .54959 .31483 L .54854 .31279 L .54733 .31085 L .54666 .30992 L .54631 .30946 L .54614 .30924 L Mfstroke .54578 .3088 m .5456 .30858 L .54522 .30814 L .54446 .3073 L .54284 .30569 L .5393 .30282 L .53545 .30039 L .53141 .29841 L .52727 .29686 L .52519 .29624 L .52311 .29571 L .52105 .29528 L .51902 .29494 L .51802 .2948 L .51702 .29469 L .51604 .29459 L .51506 .29452 L .51458 .29449 L .5141 .29447 L .51362 .29445 L .51339 .29444 L .51315 .29443 L .51291 .29442 L .51268 .29442 L .51245 .29442 L .51221 .29441 L .51198 .29441 L .51175 .29441 L .51152 .29441 L .51129 .29441 L .51106 .29442 L .51084 .29442 L .51038 .29443 L .50994 .29445 L .50949 .29447 L .50862 .29452 L .50776 .29459 L .50609 .29477 L .5045 .29501 L .50298 .2953 L .50154 .29563 L .4989 .29643 L .49659 .29737 L .49461 .2984 L .49296 .29952 L .49164 .30067 L .49063 .30184 L .49023 .30243 L .4899 .30301 L .48965 .30358 L Mistroke .48954 .30387 L .48945 .30415 L .48938 .30443 L .48932 .3047 L .48929 .30484 L .48927 .30497 L .48926 .30511 L .48924 .30524 L .48924 .30531 L .48923 .30537 L .48923 .30544 L .48923 .30551 L .48923 .30557 L .48923 .30564 L .48923 .3057 L .48923 .30577 L .48923 .30583 L .48923 .3059 L .48923 .30596 L .48924 .30602 L .48925 .30615 L .48926 .30628 L .48929 .30652 L .48934 .30677 L .48947 .30724 L .48964 .30769 L .48985 .30812 L Mfstroke P P p p .004 w .26702 .20096 m .26537 .20711 L .26404 .2132 L .26351 .21622 L .26305 .21922 L .26267 .2222 L .26237 .22516 L .26224 .22663 L .26214 .2281 L .26206 .22956 L .26199 .23102 L .26194 .23247 L .26191 .23391 L .2619 .23535 L .26191 .23679 L .26194 .23822 L .26198 .23964 L .26204 .24106 L .26212 .24247 L .26233 .24528 L .26261 .24806 L .26337 .25354 L .2644 .25893 L .26718 .26938 L .2709 .27938 L .27548 .28892 L .28692 .30654 L .30093 .32213 L .31693 .33565 L .33441 .34709 L .35287 .35649 L .37186 .36392 L .38143 .36692 L .39097 .36946 L .40045 .37157 L .40983 .37325 L .41907 .37453 L .42362 .37502 L .42813 .37542 L .43258 .37572 L .43479 .37584 L .43698 .37594 L .43916 .37601 L .44024 .37604 L .44132 .37606 L .44239 .37608 L .44346 .3761 L .44453 .37611 L .44559 .37611 L .44665 .37611 L Mistroke .44771 .3761 L .44876 .37609 L .4498 .37607 L .45188 .37602 L .45394 .37595 L .45598 .37587 L .45801 .37576 L .462 .37549 L .46592 .37515 L .46975 .37474 L .47718 .37373 L .48426 .37247 L .49099 .37098 L .50335 .36743 L .51419 .36322 L .5235 .35851 L .53128 .35345 L .53757 .34817 L .54244 .34279 L .54436 .3401 L .54596 .33742 L .54724 .33477 L .54822 .33217 L .5486 .33088 L .54891 .32961 L .54916 .32835 L .54925 .32772 L .54933 .3271 L .5494 .32649 L .54942 .32618 L .54944 .32588 L .54946 .32557 L .54948 .32527 L .54949 .32497 L .54949 .32467 L .5495 .32437 L .54949 .32407 L .54949 .32377 L .54948 .32348 L .54947 .32318 L .54945 .32289 L .54941 .3223 L .54935 .32173 L .54928 .32115 L .5491 .32002 L .54886 .31891 L .54857 .31782 L .54785 .31572 L .54694 .31371 L .54464 .30999 L Mistroke .5418 .3067 L .53852 .30383 L .53492 .30139 L .5311 .29938 L .52716 .29779 L .52518 .29714 L .52319 .29658 L .52122 .29612 L .51926 .29575 L .51733 .29546 L .51638 .29535 L .51544 .29526 L .51451 .29518 L .51404 .29515 L .51358 .29513 L .51313 .29511 L .51268 .29509 L .51245 .29508 L .51223 .29508 L .512 .29508 L .51178 .29507 L .51156 .29507 L .51134 .29507 L .51112 .29507 L .5109 .29507 L .51068 .29507 L .51046 .29508 L .51025 .29508 L .51003 .29509 L .5096 .2951 L .50918 .29512 L .50834 .29517 L .50752 .29523 L .50671 .29531 L .50515 .2955 L .50366 .29575 L .50224 .29605 L .49963 .29677 L .49733 .29763 L .49535 .29859 L .49369 .29964 L .49235 .30073 L .49129 .30185 L Mfstroke P P p p .004 w .23795 .07493 m .22358 .09238 L .21139 .1098 L .2013 .1271 L .19324 .14422 L .18712 .16108 L .18475 .16939 L .18284 .17762 L .18137 .18575 L .1808 .18977 L .18033 .19377 L .17997 .19775 L .17982 .19972 L .1797 .20169 L .17961 .20366 L .17954 .20561 L .1795 .20756 L .17948 .2095 L .17949 .21143 L .17952 .21336 L .17957 .21527 L .17965 .21718 L .17975 .21908 L .17988 .22098 L .1802 .22474 L .18062 .22847 L .18112 .23217 L .18239 .23946 L .18401 .24661 L .18595 .25361 L .19078 .26717 L .19678 .28011 L .20385 .2924 L .22082 .31496 L .2409 .33477 L .26338 .35177 L .28755 .366 L .31278 .37751 L .33847 .38641 L .35133 .38992 L .36411 .39284 L .37676 .39519 L .38922 .39698 L .39537 .39768 L .40146 .39826 L .40748 .3987 L .41046 .39888 L .41342 .39903 L .41636 .39915 L .41782 .3992 L Mistroke .41928 .39924 L .42074 .39927 L .42218 .3993 L .42363 .39932 L .42506 .39933 L .42649 .39933 L .42792 .39933 L .42934 .39932 L .43076 .39931 L .43216 .39929 L .43357 .39926 L .43636 .39918 L .43912 .39908 L .44186 .39895 L .44727 .39862 L .45257 .39819 L .45777 .39767 L .46783 .39635 L .47743 .3947 L .4952 .39052 L .51097 .38533 L .52469 .37935 L .53634 .37279 L .54596 .36583 L .5536 .35864 L .55671 .35501 L .55937 .35139 L .56158 .34779 L .56336 .34422 L .56474 .34069 L .56528 .33896 L .56573 .33724 L .56609 .33554 L .56623 .33469 L .56635 .33386 L .56645 .33302 L .56652 .3322 L .56656 .33179 L .56658 .33138 L .5666 .33097 L .56661 .33056 L .56662 .33016 L .56663 .32975 L .56663 .32935 L .56662 .32895 L .56661 .32855 L .5666 .32816 L .56658 .32776 L .56655 .32737 L .56648 .32658 L Mistroke .5664 .32581 L .56617 .32428 L .56588 .32278 L .56551 .3213 L .56457 .31845 L .56339 .31573 L .56197 .31314 L .55855 .30837 L .55445 .30418 L .54983 .30056 L .54485 .29753 L .53962 .29507 L .53429 .29314 L .53161 .29238 L .52895 .29174 L .52631 .29122 L .52371 .29082 L .52243 .29066 L .52115 .29053 L .5199 .29042 L .51927 .29038 L .51865 .29034 L .51804 .29031 L .51742 .29028 L .51712 .29027 L .51682 .29026 L .51651 .29026 L .51621 .29025 L .51591 .29025 L .51561 .29024 L .51532 .29024 L .51502 .29024 L .51473 .29024 L .51443 .29025 L .51414 .29025 L .51385 .29026 L .51327 .29027 L .51269 .29029 L .51156 .29035 L .51044 .29043 L .50935 .29053 L .50723 .29078 L .50521 .29111 L .50146 .29195 L .49813 .293 L .49522 .29422 L .49274 .29557 L Mfstroke P P p p .004 w .15031 .597 m .17023 .59904 L .18994 .60067 L .1997 .60134 L .20941 .60191 L .21905 .60238 L .22863 .60275 L .2334 .6029 L .23815 .60303 L .24288 .60313 L .24759 .60321 L .25229 .60327 L .25696 .60331 L .26162 .60332 L .26627 .60331 L .27089 .60328 L .2755 .60322 L .28008 .60315 L .28465 .60305 L .29373 .60279 L .30273 .60244 L .32048 .6015 L .33791 .60024 L .37173 .59679 L .40411 .59219 L .43498 .58653 L .49199 .57239 L .54244 .55508 L .58619 .5353 L .62322 .51369 L .65364 .49087 L .67767 .4674 L .68738 .45558 L .69561 .44378 L .70243 .43206 L .70787 .42047 L .712 .40905 L .71359 .40343 L .71487 .39786 L .71586 .39235 L .71624 .38962 L .71656 .38692 L .7168 .38422 L .71689 .38289 L .71697 .38155 L .71703 .38022 L .71708 .3789 L .7171 .37758 L .71711 .37627 L .71711 .37496 L Mistroke .71708 .37366 L .71705 .37236 L .71699 .37107 L .71683 .3685 L .71661 .36595 L .71633 .36342 L .71598 .36092 L .71511 .35599 L .71401 .35115 L .71269 .3464 L .7094 .33722 L .70533 .32847 L .70053 .32015 L .68902 .30488 L .6754 .29148 L .66016 .27997 L .64378 .27035 L .62669 .26257 L .60928 .25656 L .60057 .25418 L .59192 .25221 L .58335 .25062 L .57491 .24941 L .57075 .24894 L .56662 .24856 L .56255 .24826 L .56053 .24814 L .55853 .24804 L .55653 .24796 L .55554 .24793 L .55456 .2479 L .55357 .24788 L .55259 .24786 L .55162 .24785 L .55064 .24784 L .54967 .24784 L .54871 .24784 L .54775 .24785 L .54679 .24786 L .54583 .24787 L .54488 .24789 L .543 .24794 L .54112 .24801 L .53927 .2481 L .53561 .24833 L .53202 .24862 L .5285 .24898 L .52169 .24987 L .51519 .25099 L .50316 .25384 L Mistroke .49249 .25735 L .48321 .26141 L .47533 .26586 L .46882 .27057 L .46366 .27544 L .46155 .2779 L .45976 .28035 L .45827 .28279 L 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.24297 .42768 L .26299 .43342 L .28292 .43823 L .30266 .44213 L .32215 .44517 L .33177 .44637 L .34129 .44737 L .35071 .44818 L .36003 .44879 L .36464 .44903 L .36922 .44922 L .3715 .4493 L .37377 .44936 L .37604 .44942 L .37829 .44946 L .38054 .4495 L .38277 .44952 L .385 .44953 L .38723 .44953 L .38944 .44952 L .39164 .4495 L .39384 .44947 L .39602 .44943 L .40037 .44932 L .40467 .44916 L .41317 .44874 L .42151 .44815 L .42969 .44742 L .44555 .44553 L .4607 .44311 L .48878 .43685 L .51376 .42898 L .53556 .41984 L .55415 .40973 L .56956 .39897 L .58188 .38783 L .59126 .37655 L .59489 .37093 L .59786 .36536 L .60018 .35986 L .60111 .35714 L .60189 .35445 L .60252 .35179 L .60301 .34915 L .6032 .34785 L .60336 .34655 L Mistroke .60349 .34527 L .60353 .34463 L .60358 .34399 L .60361 .34335 L .60364 .34272 L .60365 .34209 L .60366 .34146 L .60366 .34083 L .60366 .34021 L .60364 .33959 L .60362 .33897 L .60359 .33835 L .60355 .33774 L .60345 .33652 L .60333 .33531 L .60317 .33412 L .60277 .33175 L .60226 .32944 L .60092 .32494 L .59918 .32065 L .59708 .31656 L .59465 .31268 L .59332 .31082 L .59263 .30991 L .59227 .30946 L Mfstroke .59156 .30857 m .59119 .30814 L .59045 .30727 L .58892 .30558 L .58568 .30236 L .5786 .29661 L .57091 .29177 L .56282 .28781 L .55454 .2847 L .55037 .28345 L .54622 .2824 L .54211 .28154 L .53804 .28086 L .53603 .28059 L .53404 .28036 L .53207 .28017 L .53012 .28002 L .52916 .27996 L .5282 .27991 L .52724 .27987 L .52677 .27986 L .5263 .27984 L .52583 .27983 L .52536 .27982 L .52489 .27981 L .52443 .27981 L .52396 .27981 L .5235 .27981 L .52304 .27981 L .52258 .27981 L .52213 .27982 L .52167 .27983 L .52077 .27985 L .51987 .27988 L .51898 .27992 L .51723 .28003 L .51551 .28016 L .51218 .28052 L .50899 .281 L .50595 .28158 L .50307 .28225 L .49779 .28385 L .49317 .28571 L .48922 .28779 L .48593 .29001 L .48328 .29232 L .48125 .29467 L .48046 .29584 L .47981 .297 L .47929 .29815 L Mistroke .47908 .29871 L .4789 .29928 L .47875 .29983 L .47864 .30038 L .47859 .30066 L .47855 .30093 L .47852 .3012 L .47849 .30147 L .47848 .3016 L .47847 .30173 L .47846 .30186 L .47846 .30199 L .47845 .30213 L .47845 .30226 L .47845 .30239 L .47845 .30252 L .47845 .30265 L .47846 .30277 L .47847 .3029 L .47847 .30303 L .47849 .30328 L .47852 .30354 L .47859 .30403 L .47868 .30452 L .47894 .30546 L .47928 .30637 L .4797 .30723 L Mfstroke P P p p .004 w .06311 .21892 m .06855 .23821 L .07571 .25666 L .08444 .27425 L .0946 .29093 L .11871 .32151 L .14697 .34826 L .17838 .37115 L .212 .39021 L .24695 .40554 L .28242 .4173 L .30014 .4219 L .31772 .4257 L .3351 .4287 L .3437 .42993 L .35221 .43097 L .36064 .43183 L .36898 .43251 L .37312 .43279 L .37723 .43303 L .38131 .43323 L .38536 .43338 L .38738 .43344 L .38939 .4335 L .39139 .43354 L .39338 .43357 L .39537 .4336 L .39735 .43361 L .39932 .43361 L .40129 .43361 L .40324 .43359 L .40519 .43357 L .40713 .43354 L .40906 .43349 L .41291 .43338 L .41671 .43323 L .42423 .43283 L .4316 .43229 L .44591 .43082 L .45961 .42887 L .47267 .42647 L .49678 .42047 L .51812 .41313 L .53662 .40475 L .55227 .3956 L .56513 .38593 L .57528 .37599 L .57939 .37099 L .58286 .366 L .58574 .36104 L Mistroke .58803 .35613 L .58977 .3513 L .59044 .34891 L .59098 .34656 L .5914 .34423 L .59156 .34307 L .59169 .34193 L .59179 .34079 L .59183 .34022 L .59186 .33966 L .59189 .3391 L .5919 .33854 L .59191 .33798 L .59192 .33742 L .59191 .33687 L .5919 .33632 L .59189 .33577 L .59186 .33522 L .59179 .33413 L .5917 .33305 L .59158 .33199 L .59143 .33093 L .59106 .32884 L .59059 .32679 L .59003 .32478 L .58863 .3209 L .5869 .3172 L .58486 .31368 L .57998 .30722 L .57421 .30155 L .56775 .29669 L .56082 .29262 L .55358 .28933 L .54621 .28679 L .54252 .28579 L .53886 .28495 L .53523 .28429 L .53166 .28378 L .5299 .28358 L .52815 .28342 L .52643 .28329 L .52558 .28324 L .52473 .2832 L .52388 .28317 L .52346 .28315 L .52305 .28314 L .52263 .28313 L .52221 .28313 L .5218 .28312 L .52139 .28312 L Mistroke .52098 .28312 L .52057 .28312 L .52016 .28312 L .51976 .28312 L .51936 .28313 L .51895 .28314 L .51815 .28316 L .51736 .28319 L .51658 .28323 L .51503 .28333 L .51351 .28345 L .51202 .2836 L .50914 .28398 L .50639 .28446 L .5013 .28567 L .49679 .28716 L .49286 .28887 L .48953 .29076 L .48678 .29276 L .4846 .29482 L .48371 .29586 L .48295 .2969 L .48232 .29793 L .48181 .29895 L .48142 .29996 L .48114 .30095 L Mfstroke P P p p .004 w .03404 .0929 m .03073 .10521 L .02809 .11739 L .02701 .12342 L .0261 .12942 L .02534 .13538 L .02473 .1413 L .02449 .14425 L .02428 .14718 L .02411 .15011 L .02398 .15302 L .02389 .15592 L .02383 .15881 L .02381 .16169 L .02383 .16456 L .02388 .16742 L .02397 .17027 L .02409 .1731 L .02425 .17592 L .02467 .18153 L .02523 .1871 L .02675 .19807 L .02879 .20884 L .03437 .22974 L .04181 .24975 L .05097 .26883 L .07385 .30406 L .10185 .33525 L .13387 .36229 L .16883 .38517 L .20575 .40397 L .24373 .41882 L .26285 .42482 L .28194 .42991 L .30091 .43412 L .31966 .43749 L .33814 .44004 L .34724 .44102 L .35626 .44181 L .36516 .44242 L .36957 .44266 L .37396 .44286 L .37831 .44301 L .38048 .44306 L .38263 .44311 L .38478 .44315 L .38693 .44318 L .38906 .44319 L .39118 .4432 L .3933 .4432 L Mistroke .39541 .44318 L .39751 .44316 L .3996 .44312 L .40376 .44303 L .40788 .44289 L .41197 .44271 L .41602 .4425 L .424 .44196 L .43183 .44128 L .43951 .44047 L .45436 .43844 L .46853 .43592 L .48198 .43295 L .50669 .42584 L .52838 .41742 L .54699 .408 L .56256 .39788 L .57515 .38731 L .58488 .37656 L .58873 .37118 L .59192 .36583 L .59448 .36053 L .59644 .35531 L .5972 .35274 L .59783 .35019 L .59831 .34768 L .5985 .34643 L .59866 .34519 L .59879 .34396 L .59884 .34334 L .59889 .34273 L .59892 .34213 L .59895 .34152 L .59897 .34092 L .59898 .34032 L .59899 .33972 L .59899 .33912 L .59898 .33853 L .59896 .33793 L .59894 .33734 L .5989 .33676 L .59882 .33559 L .5987 .33443 L .59856 .33329 L .59819 .33103 L .59772 .32881 L .59714 .32663 L .5957 .32242 L .59388 .3184 L .58929 .31097 L Mistroke .58359 .30438 L .57704 .29864 L .56984 .29377 L .5622 .28975 L .55433 .28656 L .55035 .28526 L .54638 .28415 L .54243 .28322 L .53852 .28248 L .53466 .28191 L .53276 .28168 L .53087 .28149 L .52901 .28135 L .52809 .28129 L .52717 .28124 L .52626 .2812 L .52535 .28117 L .5249 .28115 L .52445 .28114 L .52401 .28113 L .52356 .28113 L .52312 .28112 L .52268 .28112 L .52223 .28112 L .5218 .28113 L .52136 .28113 L .52092 .28114 L .52049 .28115 L .52006 .28116 L .5192 .28119 L .51835 .28122 L .51668 .28132 L .51503 .28144 L .51342 .2816 L .51029 .28199 L .50731 .28249 L .50447 .28308 L .49925 .28452 L .49466 .28624 L .49071 .28817 L .48739 .29026 L .48469 .29245 L .48259 .29468 L Mfstroke P P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = text; inactive; preserveAspect] The trajectories spiral into the origin; {0,0} is stable. ;[s] 3:0,0;41,1;46,0;58,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = text; inactive; preserveAspect] The case h = 0.0: ;[s] 3:0,1;9,2;16,1;18,-1; 3:0,13,9,Times,2,12,0,0,0;2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] PlotVectorField[{y,-x},{x,-2,2},{y,-2,2}, ScaleFunction->(Log[50#+1]&), Frame->True] ;[s] 3:0,0;75,1;76,0;89,-1; 2:2,12,10,Courier,1,12,0,0,0;1,12,9,Times,0,12,0,0,0; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 282] %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.216251 0.5 0.216251 [ [(-2)] .0675 0 0 2 Msboxa [(-1)] .28375 0 0 2 Msboxa [(0)] .5 0 0 2 Msboxa [(1)] .71625 0 0 2 Msboxa [(2)] .9325 0 0 2 Msboxa [(-2)] -0.0125 .0675 1 0 Msboxa [(-1)] -0.0125 .28375 1 0 Msboxa [(0)] -0.0125 .5 1 0 Msboxa [(1)] -0.0125 .71625 1 0 Msboxa [(2)] 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p .001 w 0 .7595 m .00375 .7595 L s P p .001 w 0 .80275 m .00375 .80275 L s P p .001 w 0 .846 m .00375 .846 L s P p .001 w 0 .88925 m .00375 .88925 L s P p .001 w 0 .02425 m .00375 .02425 L s P p .001 w 0 .97575 m .00375 .97575 L s P p .002 w 0 0 m 0 1 L s P P p p .002 w .0675 .99375 m .0675 1 L s P p .002 w .28375 .99375 m .28375 1 L s P p .002 w .5 .99375 m .5 1 L s P p .002 w .71625 .99375 m .71625 1 L s P p .002 w .9325 .99375 m .9325 1 L s P p .001 w .11075 .99625 m .11075 1 L s P p .001 w .154 .99625 m .154 1 L s P p .001 w .19725 .99625 m .19725 1 L s P p .001 w .2405 .99625 m .2405 1 L s P p .001 w .327 .99625 m .327 1 L s P p .001 w .37025 .99625 m .37025 1 L s P p .001 w .4135 .99625 m .4135 1 L s P p .001 w .45675 .99625 m .45675 1 L s P p .001 w .54325 .99625 m .54325 1 L s P p .001 w .5865 .99625 m .5865 1 L s P p .001 w .62975 .99625 m .62975 1 L s P p .001 w .673 .99625 m .673 1 L s P p .001 w .7595 .99625 m .7595 1 L s P p .001 w .80275 .99625 m .80275 1 L s P p .001 w 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.45333 L .0254 .46109 L .02483 .46886 L .02438 .47663 L .02421 .48053 L .02406 .48442 L .02395 .48831 L .02387 .49221 L .02383 .4961 L s P P p p .004 w .14287 .50292 m .1429 .50584 L .14296 .50876 L .14305 .51169 L .14316 .51461 L .14329 .51752 L .14362 .52336 L .14405 .52919 L .14458 .53501 L .14591 .54662 L .14763 .55818 L .14972 .56968 L .15503 .59244 L .16181 .6148 L .17004 .63667 L .19071 .67857 L .21666 .71741 L .24746 .75254 L .28259 .78334 L .32143 .80929 L .34204 .82031 L .36333 .82996 L .3852 .83819 L .40756 .84497 L .43032 .85028 L .44182 .85237 L .45338 .85409 L .46499 .85542 L .47081 .85595 L .47664 .85638 L .48248 .85671 L .48539 .85684 L .48831 .85695 L .49124 .85704 L .49416 .8571 L .49708 .85713 L .5 .85714 L .50292 .85713 L .50584 .8571 L .50876 .85704 L .51169 .85695 L .51752 .85671 L .52336 .85638 L .52919 .85595 L .53501 .85542 L .54662 .85409 L .55818 .85237 L .56968 .85028 L .59244 .84497 L .6148 .83819 L Mistroke .63667 .82996 L .67857 .80929 L .71741 .78334 L .75254 .75254 L .78334 .71741 L .80929 .67857 L .82031 .65796 L .82996 .63667 L .83819 .6148 L .84497 .59244 L .85028 .56968 L .85237 .55818 L .85409 .54662 L .85542 .53501 L .85595 .52919 L .85638 .52336 L .85671 .51752 L .85684 .51461 L .85695 .51169 L .85704 .50876 L .8571 .50584 L .85713 .50292 L .85714 .5 L .85713 .49708 L .8571 .49416 L .85704 .49124 L .85695 .48831 L .85684 .48539 L .85671 .48248 L .85638 .47664 L .85595 .47081 L .85542 .46499 L .85409 .45338 L .85237 .44182 L .85028 .43032 L .84497 .40756 L .83819 .3852 L .82996 .36333 L .80929 .32143 L .78334 .28259 L .75254 .24746 L .71741 .21666 L .67857 .19071 L .65796 .17969 L .63667 .17004 L .6148 .16181 L .59244 .15503 L .56968 .14972 L .55818 .14763 L .54662 .14591 L Mistroke .53501 .14458 L .52919 .14405 L .52336 .14362 L .51752 .14329 L .51461 .14316 L .51169 .14305 L .50876 .14296 L .50584 .1429 L .50292 .14287 L Mfstroke .49708 .14287 m .49416 .1429 L .49124 .14296 L .48831 .14305 L .48248 .14329 L .47664 .14362 L .47081 .14405 L .46499 .14458 L .45338 .14591 L .44182 .14763 L .43032 .14972 L .40756 .15503 L .3852 .16181 L .36333 .17004 L .32143 .19071 L .28259 .21666 L .24746 .24746 L .21666 .28259 L .19071 .32143 L .17969 .34204 L .17004 .36333 L .16181 .3852 L .15503 .40756 L .14972 .43032 L .14763 .44182 L .14591 .45338 L .14458 .46499 L .14405 .47081 L .14362 .47664 L .14329 .48248 L .14316 .48539 L .14305 .48831 L .14296 .49124 L .1429 .49416 L .14287 .49708 L s P P p p .004 w .26191 .50195 m .26194 .5039 L .26198 .50584 L .26203 .50779 L .2621 .50974 L .26219 .51168 L .26241 .51557 L .2627 .51946 L .26305 .52334 L .26394 .53108 L .26508 .53878 L .26648 .54645 L .27002 .56162 L .27454 .57653 L .28003 .59112 L .2938 .61905 L .31111 .64494 L .33164 .66836 L .35506 .68889 L .38095 .7062 L .39469 .71354 L .40888 .71997 L .42347 .72546 L .43838 .72998 L .45355 .73352 L .46122 .73492 L .46892 .73606 L .47666 .73695 L .48054 .7373 L .48443 .73759 L .48832 .73781 L .49026 .7379 L .49221 .73797 L .49416 .73802 L .4961 .73806 L .49805 .73809 L .5 .7381 L .50195 .73809 L .5039 .73806 L .50584 .73802 L .50779 .73797 L .51168 .73781 L .51557 .73759 L .51946 .7373 L .52334 .73695 L .53108 .73606 L .53878 .73492 L .54645 .73352 L .56162 .72998 L .57653 .72546 L Mistroke .59112 .71997 L .61905 .7062 L .64494 .68889 L .66836 .66836 L .68889 .64494 L .7062 .61905 L .71354 .60531 L .71997 .59112 L .72546 .57653 L .72998 .56162 L .73352 .54645 L .73492 .53878 L .73606 .53108 L .73695 .52334 L .7373 .51946 L .73759 .51557 L .73781 .51168 L .7379 .50974 L .73797 .50779 L .73802 .50584 L .73806 .5039 L .73809 .50195 L .7381 .5 L .73809 .49805 L .73806 .4961 L .73802 .49416 L .73797 .49221 L .7379 .49026 L .73781 .48832 L .73759 .48443 L .7373 .48054 L .73695 .47666 L .73606 .46892 L .73492 .46122 L .73352 .45355 L .72998 .43838 L .72546 .42347 L .71997 .40888 L .7062 .38095 L .68889 .35506 L .66836 .33164 L .64494 .31111 L .61905 .2938 L .60531 .28646 L .59112 .28003 L .57653 .27454 L .56162 .27002 L .54645 .26648 L .53878 .26508 L .53108 .26394 L Mistroke .52334 .26305 L .51946 .2627 L .51557 .26241 L .51168 .26219 L .50974 .2621 L .50779 .26203 L .50584 .26198 L .5039 .26194 L .50195 .26191 L Mfstroke .49805 .26191 m .4961 .26194 L .49416 .26198 L .49221 .26203 L .48832 .26219 L .48443 .26241 L .48054 .2627 L .47666 .26305 L .46892 .26394 L .46122 .26508 L .45355 .26648 L .43838 .27002 L .42347 .27454 L .40888 .28003 L .38095 .2938 L .35506 .31111 L .33164 .33164 L .31111 .35506 L .2938 .38095 L .28646 .39469 L .28003 .40888 L .27454 .42347 L .27002 .43838 L .26648 .45355 L .26508 .46122 L .26394 .46892 L .26305 .47666 L .2627 .48054 L .26241 .48443 L .26219 .48832 L .2621 .49026 L .26203 .49221 L .26198 .49416 L .26194 .4961 L .26191 .49805 L s P P p p .004 w .38096 .50097 m .38097 .50195 L .38099 .50292 L .38102 .5039 L .38105 .50487 L .3811 .50584 L .38121 .50779 L .38135 .50973 L .38153 .51167 L .38197 .51554 L .38254 .51939 L .38324 .52323 L .38501 .53081 L .38727 .53827 L .39001 .54556 L .3969 .55952 L .40555 .57247 L .41582 .58418 L .42753 .59445 L .44048 .6031 L .44735 .60677 L .45444 .60999 L .46173 .61273 L .46919 .61499 L .47677 .61676 L .48061 .61746 L .48446 .61803 L .48833 .61847 L .49027 .61865 L .49221 .61879 L .49416 .6189 L .49513 .61895 L .4961 .61898 L .49708 .61901 L .49805 .61903 L .49903 .61904 L .5 .61905 L .50097 .61904 L .50195 .61903 L .50292 .61901 L .5039 .61898 L .50584 .6189 L .50779 .61879 L .50973 .61865 L .51167 .61847 L .51554 .61803 L .51939 .61746 L .52323 .61676 L .53081 .61499 L .53827 .61273 L Mistroke .54556 .60999 L .55952 .6031 L .57247 .59445 L .58418 .58418 L .59445 .57247 L .6031 .55952 L .60677 .55265 L .60999 .54556 L .61273 .53827 L .61499 .53081 L .61676 .52323 L .61746 .51939 L .61803 .51554 L .61847 .51167 L .61865 .50973 L .61879 .50779 L .6189 .50584 L .61895 .50487 L .61898 .5039 L .61901 .50292 L .61903 .50195 L .61904 .50097 L .61905 .5 L .61904 .49903 L .61903 .49805 L .61901 .49708 L .61898 .4961 L .61895 .49513 L .6189 .49416 L .61879 .49221 L .61865 .49027 L .61847 .48833 L .61803 .48446 L .61746 .48061 L .61676 .47677 L .61499 .46919 L .61273 .46173 L .60999 .45444 L .6031 .44048 L .59445 .42753 L .58418 .41582 L .57247 .40555 L .55952 .3969 L .55265 .39323 L .54556 .39001 L .53827 .38727 L .53081 .38501 L .52323 .38324 L .51939 .38254 L .51554 .38197 L Mistroke .51167 .38153 L .50973 .38135 L .50779 .38121 L .50584 .3811 L .50487 .38105 L .5039 .38102 L .50292 .38099 L .50195 .38097 L .50097 .38096 L Mfstroke .49903 .38096 m .49805 .38097 L .49708 .38099 L .4961 .38102 L .49416 .3811 L .49221 .38121 L .49027 .38135 L .48833 .38153 L .48446 .38197 L .48061 .38254 L .47677 .38324 L .46919 .38501 L .46173 .38727 L .45444 .39001 L .44048 .3969 L .42753 .40555 L .41582 .41582 L .40555 .42753 L .3969 .44048 L .39323 .44735 L .39001 .45444 L .38727 .46173 L .38501 .46919 L .38324 .47677 L .38254 .48061 L .38197 .48446 L .38153 .48833 L .38135 .49027 L .38121 .49221 L .3811 .49416 L .38105 .49513 L .38102 .4961 L .38099 .49708 L .38097 .49805 L .38096 .49903 L s P P p p P P p p .004 w .61904 .49903 m .61903 .49805 L .61901 .49708 L .61898 .4961 L .61895 .49513 L .6189 .49416 L .61879 .49221 L .61865 .49027 L .61847 .48833 L .61803 .48446 L .61746 .48061 L .61676 .47677 L .61499 .46919 L .61273 .46173 L .60999 .45444 L .6031 .44048 L .59445 .42753 L .58418 .41582 L .57247 .40555 L .55952 .3969 L .55265 .39323 L .54556 .39001 L .53827 .38727 L .53081 .38501 L .52323 .38324 L .51939 .38254 L .51554 .38197 L .51167 .38153 L .50973 .38135 L .50779 .38121 L .50584 .3811 L .50487 .38105 L .5039 .38102 L .50292 .38099 L .50195 .38097 L .50097 .38096 L .5 .38095 L .49903 .38096 L .49805 .38097 L .49708 .38099 L .4961 .38102 L .49416 .3811 L .49221 .38121 L .49027 .38135 L .48833 .38153 L .48446 .38197 L .48061 .38254 L .47677 .38324 L .46919 .38501 L .46173 .38727 L Mistroke .45444 .39001 L .44048 .3969 L .42753 .40555 L .41582 .41582 L .40555 .42753 L .3969 .44048 L .39323 .44735 L .39001 .45444 L .38727 .46173 L .38501 .46919 L .38324 .47677 L .38254 .48061 L .38197 .48446 L .38153 .48833 L .38135 .49027 L .38121 .49221 L .3811 .49416 L .38105 .49513 L .38102 .4961 L .38099 .49708 L .38097 .49805 L .38096 .49903 L .38095 .5 L .38096 .50097 L .38097 .50195 L .38099 .50292 L .38102 .5039 L .38105 .50487 L .3811 .50584 L .38121 .50779 L .38135 .50973 L .38153 .51167 L .38197 .51554 L .38254 .51939 L .38324 .52323 L .38501 .53081 L .38727 .53827 L .39001 .54556 L .3969 .55952 L .40555 .57247 L .41582 .58418 L .42753 .59445 L .44048 .6031 L .44735 .60677 L .45444 .60999 L .46173 .61273 L .46919 .61499 L .47677 .61676 L .48061 .61746 L .48446 .61803 L Mistroke .48833 .61847 L .49027 .61865 L .49221 .61879 L .49416 .6189 L .49513 .61895 L .4961 .61898 L .49708 .61901 L .49805 .61903 L .49903 .61904 L Mfstroke .50097 .61904 m .50195 .61903 L .50292 .61901 L .5039 .61898 L .50584 .6189 L .50779 .61879 L .50973 .61865 L .51167 .61847 L .51554 .61803 L .51939 .61746 L .52323 .61676 L .53081 .61499 L .53827 .61273 L .54556 .60999 L .55952 .6031 L .57247 .59445 L .58418 .58418 L .59445 .57247 L .6031 .55952 L .60677 .55265 L .60999 .54556 L .61273 .53827 L .61499 .53081 L .61676 .52323 L .61746 .51939 L .61803 .51554 L .61847 .51167 L .61865 .50973 L .61879 .50779 L .6189 .50584 L .61895 .50487 L .61898 .5039 L .61901 .50292 L .61903 .50195 L .61904 .50097 L s P P p p .004 w .73809 .49805 m .73806 .4961 L .73802 .49416 L .73797 .49221 L .7379 .49026 L .73781 .48832 L .73759 .48443 L .7373 .48054 L .73695 .47666 L .73606 .46892 L .73492 .46122 L .73352 .45355 L .72998 .43838 L .72546 .42347 L .71997 .40888 L .7062 .38095 L .68889 .35506 L .66836 .33164 L .64494 .31111 L .61905 .2938 L .60531 .28646 L .59112 .28003 L .57653 .27454 L .56162 .27002 L .54645 .26648 L .53878 .26508 L .53108 .26394 L .52334 .26305 L .51946 .2627 L .51557 .26241 L .51168 .26219 L .50974 .2621 L .50779 .26203 L .50584 .26198 L .5039 .26194 L .50195 .26191 L .5 .2619 L .49805 .26191 L .4961 .26194 L .49416 .26198 L .49221 .26203 L .48832 .26219 L .48443 .26241 L .48054 .2627 L .47666 .26305 L .46892 .26394 L .46122 .26508 L .45355 .26648 L .43838 .27002 L .42347 .27454 L Mistroke .40888 .28003 L .38095 .2938 L .35506 .31111 L .33164 .33164 L .31111 .35506 L .2938 .38095 L .28646 .39469 L .28003 .40888 L .27454 .42347 L .27002 .43838 L .26648 .45355 L .26508 .46122 L .26394 .46892 L .26305 .47666 L .2627 .48054 L .26241 .48443 L .26219 .48832 L .2621 .49026 L .26203 .49221 L .26198 .49416 L .26194 .4961 L .26191 .49805 L .2619 .5 L .26191 .50195 L .26194 .5039 L .26198 .50584 L .26203 .50779 L .2621 .50974 L .26219 .51168 L .26241 .51557 L .2627 .51946 L .26305 .52334 L .26394 .53108 L .26508 .53878 L .26648 .54645 L .27002 .56162 L .27454 .57653 L .28003 .59112 L .2938 .61905 L .31111 .64494 L .33164 .66836 L .35506 .68889 L .38095 .7062 L .39469 .71354 L .40888 .71997 L .42347 .72546 L .43838 .72998 L .45355 .73352 L .46122 .73492 L .46892 .73606 L Mistroke .47666 .73695 L .48054 .7373 L .48443 .73759 L .48832 .73781 L .49026 .7379 L .49221 .73797 L .49416 .73802 L .4961 .73806 L .49805 .73809 L Mfstroke .50195 .73809 m .5039 .73806 L .50584 .73802 L .50779 .73797 L .51168 .73781 L .51557 .73759 L .51946 .7373 L .52334 .73695 L .53108 .73606 L .53878 .73492 L .54645 .73352 L .56162 .72998 L .57653 .72546 L .59112 .71997 L .61905 .7062 L .64494 .68889 L .66836 .66836 L .68889 .64494 L .7062 .61905 L .71354 .60531 L .71997 .59112 L .72546 .57653 L .72998 .56162 L .73352 .54645 L .73492 .53878 L .73606 .53108 L .73695 .52334 L .7373 .51946 L .73759 .51557 L .73781 .51168 L .7379 .50974 L .73797 .50779 L .73802 .50584 L .73806 .5039 L .73809 .50195 L s P P p p .004 w .85713 .49708 m .8571 .49416 L .85704 .49124 L .85695 .48831 L .85684 .48539 L .85671 .48248 L .85638 .47664 L .85595 .47081 L .85542 .46499 L .85409 .45338 L .85237 .44182 L .85028 .43032 L .84497 .40756 L .83819 .3852 L .82996 .36333 L .80929 .32143 L .78334 .28259 L .75254 .24746 L .71741 .21666 L .67857 .19071 L .65796 .17969 L .63667 .17004 L .6148 .16181 L .59244 .15503 L .56968 .14972 L .55818 .14763 L .54662 .14591 L .53501 .14458 L .52919 .14405 L .52336 .14362 L .51752 .14329 L .51461 .14316 L .51169 .14305 L .50876 .14296 L .50584 .1429 L .50292 .14287 L .5 .14286 L .49708 .14287 L .49416 .1429 L .49124 .14296 L .48831 .14305 L .48248 .14329 L .47664 .14362 L .47081 .14405 L .46499 .14458 L .45338 .14591 L .44182 .14763 L .43032 .14972 L .40756 .15503 L .3852 .16181 L Mistroke .36333 .17004 L .32143 .19071 L .28259 .21666 L .24746 .24746 L .21666 .28259 L .19071 .32143 L .17969 .34204 L .17004 .36333 L .16181 .3852 L .15503 .40756 L .14972 .43032 L .14763 .44182 L .14591 .45338 L .14458 .46499 L .14405 .47081 L .14362 .47664 L .14329 .48248 L .14316 .48539 L .14305 .48831 L .14296 .49124 L .1429 .49416 L .14287 .49708 L .14286 .5 L .14287 .50292 L .1429 .50584 L .14296 .50876 L .14305 .51169 L .14316 .51461 L .14329 .51752 L .14362 .52336 L .14405 .52919 L .14458 .53501 L .14591 .54662 L .14763 .55818 L .14972 .56968 L .15503 .59244 L .16181 .6148 L .17004 .63667 L .19071 .67857 L .21666 .71741 L .24746 .75254 L .28259 .78334 L .32143 .80929 L .34204 .82031 L .36333 .82996 L .3852 .83819 L .40756 .84497 L .43032 .85028 L .44182 .85237 L .45338 .85409 L Mistroke .46499 .85542 L .47081 .85595 L .47664 .85638 L .48248 .85671 L .48539 .85684 L .48831 .85695 L .49124 .85704 L .49416 .8571 L .49708 .85713 L Mfstroke .50292 .85713 m .50584 .8571 L .50876 .85704 L .51169 .85695 L .51752 .85671 L .52336 .85638 L .52919 .85595 L .53501 .85542 L .54662 .85409 L .55818 .85237 L .56968 .85028 L .59244 .84497 L .6148 .83819 L .63667 .82996 L .67857 .80929 L .71741 .78334 L .75254 .75254 L .78334 .71741 L .80929 .67857 L .82031 .65796 L .82996 .63667 L .83819 .6148 L .84497 .59244 L .85028 .56968 L .85237 .55818 L .85409 .54662 L .85542 .53501 L .85595 .52919 L .85638 .52336 L .85671 .51752 L .85684 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.31777 L .04908 .34693 L .04004 .37675 L .03296 .4071 L .03017 .42243 L .02788 .43784 L .0261 .45333 L .0254 .46109 L .02483 .46886 L .02438 .47663 L .02421 .48053 L .02406 .48442 L .02395 .48831 L .02387 .49221 L .02383 .4961 L .02381 .5 L .02383 .5039 L .02387 .50779 L .02395 .51169 L .02406 .51558 L .02421 .51947 L .02438 .52337 L .02483 .53114 L .0254 .53891 L .0261 .54667 L .02788 .56216 L .03017 .57757 L .03296 .5929 L .04004 .62325 L .04908 .65307 L .06006 .68223 L .08761 .7381 L .12221 .78989 L .16328 .83672 L .21011 .87779 L .2619 .91239 L .28939 .92708 L .31777 .93994 L .34693 .95092 L .37675 .95996 L .4071 .96704 L .42243 .96983 L .43784 .97212 L Mistroke .45333 .9739 L .46109 .9746 L .46886 .97517 L .47663 .97562 L .48053 .97579 L .48442 .97594 L .48831 .97605 L .49221 .97613 L .4961 .97617 L Mfstroke .5039 .97617 m .50779 .97613 L .51169 .97605 L .51558 .97594 L .52337 .97562 L .53114 .97517 L .53891 .9746 L .54667 .9739 L .56216 .97212 L .57757 .96983 L .5929 .96704 L 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.32564 L .77918 .30307 L .7591 .28327 L .73662 .26626 L .71245 .25204 L .68722 .24053 L .66153 .23163 L .64867 .22811 L .63589 .22519 L .62324 .22285 L .61078 .22105 L .60463 .22035 L .59854 .21978 L .59252 .21933 L .58954 .21915 L .58658 .21901 L .58364 .21889 L .58218 .21884 L Mistroke .58072 .2188 L .57926 .21876 L .57782 .21874 L .57637 .21872 L .57494 .21871 L .57351 .2187 L .57208 .2187 L .57066 .21871 L .56924 .21872 L .56784 .21875 L .56643 .21877 L .56364 .21885 L .56088 .21895 L .55814 .21908 L .55273 .21941 L .54743 .21984 L .54223 .22037 L .53217 .22168 L .52257 .22333 L .5048 .22752 L .48903 .23271 L .47531 .23868 L .46366 .24525 L .45404 .25221 L .4464 .25939 L .44329 .26302 L .44063 .26664 L .43842 .27025 L .43664 .27382 L .43526 .27734 L .43472 .27908 L .43427 .2808 L .43391 .2825 L .43377 .28334 L .43365 .28418 L .43355 .28501 L .43348 .28584 L .43344 .28625 L .43342 .28666 L .4334 .28706 L .43339 .28747 L .43338 .28788 L .43337 .28828 L .43337 .28868 L .43338 .28908 L .43339 .28948 L .4334 .28988 L .43342 .29027 L .43345 .29067 L .43352 .29145 L Mistroke .4336 .29223 L .43383 .29376 L .43412 .29526 L .43449 .29673 L .43543 .29958 L .43661 .30231 L .43803 .3049 L .44145 .30966 L .44555 .31386 L .45017 .31747 L .45515 .3205 L .46038 .32297 L .46571 .32489 L .46839 .32565 L .47105 .32629 L .47369 .32681 L .47629 .32722 L .47757 .32738 L .47885 .32751 L .4801 .32761 L .48073 .32766 L .48135 .3277 L .48196 .32773 L .48258 .32775 L .48288 .32776 L .48318 .32777 L .48349 .32778 L .48379 .32778 L .48409 .32779 L .48439 .32779 L .48468 .32779 L .48498 .32779 L .48527 .32779 L .48557 .32779 L .48586 .32778 L .48615 .32778 L .48673 .32776 L .48731 .32774 L .48844 .32768 L .48956 .3276 L .49065 .32751 L .49277 .32725 L .49479 .32693 L .49854 .32609 L .50187 .32503 L .50478 .32381 L .50726 .32246 L Mfstroke P P p p .004 w .73298 .41708 m .73463 .41092 L .73596 .40483 L .73649 .40181 L .73695 .39881 L .73733 .39583 L .73763 .39287 L .73776 .3914 L .73786 .38993 L .73794 .38847 L .73801 .38702 L .73806 .38556 L .73809 .38412 L .7381 .38268 L .73809 .38124 L .73806 .37982 L .73802 .37839 L .73796 .37698 L .73788 .37556 L .73767 .37276 L .73739 .36998 L .73663 .36449 L .7356 .35911 L .73282 .34865 L .7291 .33865 L .72452 .32911 L .71308 .31149 L .69907 .2959 L .68307 .28238 L .66559 .27094 L .64713 .26154 L .62814 .25412 L .61857 .25111 L .60903 .24857 L .59955 .24646 L .59017 .24478 L .58093 .24351 L .57638 .24302 L .57187 .24262 L .56742 .24231 L .56521 .24219 L .56302 .2421 L .56084 .24202 L .55976 .24199 L .55868 .24197 L .55761 .24195 L .55654 .24194 L .55547 .24193 L .55441 .24193 L .55335 .24193 L Mistroke .55229 .24193 L .55124 .24195 L .5502 .24196 L .54812 .24201 L .54606 .24208 L .54402 .24217 L .54199 .24228 L .538 .24254 L .53408 .24288 L .53025 .24329 L .52282 .24431 L .51574 .24557 L .50901 .24705 L .49665 .25061 L .48581 .25482 L .4765 .25952 L .46872 .26459 L .46243 .26987 L .45756 .27525 L .45564 .27794 L .45404 .28061 L .45276 .28326 L .45178 .28587 L .4514 .28716 L .45109 .28843 L .45084 .28969 L .45075 .29031 L .45067 .29093 L .4506 .29155 L .45058 .29185 L .45056 .29216 L .45054 .29246 L .45052 .29277 L .45051 .29307 L .45051 .29337 L .4505 .29367 L .45051 .29397 L .45051 .29426 L .45052 .29456 L .45053 .29485 L .45055 .29515 L .45059 .29573 L .45065 .29631 L .45072 .29688 L .4509 .29801 L .45114 .29912 L .45143 .30021 L .45215 .30231 L .45306 .30432 L .45536 .30804 L Mistroke .4582 .31134 L .46148 .31421 L .46508 .31664 L .4689 .31865 L .47284 .32025 L .47482 .3209 L .47681 .32145 L .47878 .32191 L .48074 .32229 L .48267 .32257 L .48362 .32269 L .48456 .32278 L .48549 .32285 L .48596 .32288 L .48642 .32291 L .48687 .32293 L .48732 .32294 L .48755 .32295 L .48777 .32295 L .488 .32296 L .48822 .32296 L .48844 .32296 L .48866 .32296 L .48888 .32296 L .4891 .32296 L .48932 .32296 L .48954 .32296 L .48975 .32295 L .48997 .32295 L .4904 .32293 L .49082 .32291 L .49166 .32287 L .49248 .3228 L .49329 .32273 L .49485 .32253 L .49634 .32228 L .49776 .32198 L .50037 .32127 L .50267 .32041 L .50465 .31944 L .50631 .3184 L .50765 .3173 L .50871 .31619 L Mfstroke P P p p .004 w .70391 .29105 m .68666 .27751 L .66801 .26614 L .64848 .25689 L .62851 .24968 L .6185 .24681 L .60854 .24441 L .59867 .24246 L .58893 .24094 L .58412 .24034 L .57935 .23984 L .57464 .23944 L .56999 .23913 L .56768 .23901 L .56539 .23892 L .56425 .23888 L .56311 .23884 L .56198 .23882 L .56085 .23879 L .55973 .23878 L .55861 .23876 L .5575 .23876 L .55639 .23876 L .55528 .23876 L .55418 .23877 L .55308 .23879 L .55199 .23881 L .54982 .23887 L .54766 .23894 L .54341 .23916 L .53924 .23945 L .53515 .23981 L .52723 .24076 L .51965 .24197 L .50561 .2451 L .49312 .24904 L .48222 .25361 L .47293 .25866 L .46522 .26404 L .45906 .26961 L .45437 .27525 L .45255 .27806 L .45107 .28085 L .44991 .2836 L .44945 .28496 L .44906 .2863 L .44874 .28763 L .4485 .28895 L .4484 .2896 L .44832 .29025 L Mistroke .44826 .29089 L .44823 .29121 L .44821 .29153 L .4482 .29185 L .44818 .29217 L .44817 .29248 L .44817 .2928 L .44817 .29311 L .44817 .29342 L .44818 .29373 L .44819 .29404 L .44821 .29435 L .44822 .29465 L .44827 .29526 L .44834 .29587 L .44842 .29647 L .44862 .29765 L .44887 .29881 L .44954 .30105 L .45041 .3032 L .45146 .30525 L .45267 .30719 L .45334 .30812 L .45369 .30857 L .45386 .30879 L Mfstroke .45422 .30924 m .4544 .30946 L .45478 .30989 L .45554 .31074 L .45716 .31234 L .4607 .31522 L .46455 .31764 L .46859 .31962 L .47273 .32118 L .47481 .3218 L .47689 .32232 L .47895 .32275 L .48098 .32309 L .48198 .32323 L .48298 .32334 L .48396 .32344 L .48494 .32351 L .48542 .32354 L .4859 .32357 L .48638 .32359 L .48661 .3236 L .48685 .3236 L .48709 .32361 L .48732 .32361 L .48755 .32362 L .48779 .32362 L .48802 .32362 L .48825 .32362 L .48848 .32362 L .48871 .32362 L .48894 .32362 L .48916 .32361 L .48962 .3236 L .49006 .32358 L .49051 .32357 L .49138 .32351 L .49224 .32345 L .49391 .32326 L .4955 .32303 L .49702 .32274 L .49846 .3224 L .5011 .3216 L .50341 .32067 L .50539 .31963 L .50704 .31852 L .50836 .31736 L .50937 .31619 L .50977 .31561 L .5101 .31503 L .51035 .31445 L Mistroke .51046 .31417 L .51055 .31389 L .51062 .31361 L .51068 .31333 L .51071 .3132 L .51073 .31306 L .51074 .31293 L .51076 .31279 L .51076 .31273 L .51077 .31266 L .51077 .31259 L .51077 .31253 L .51077 .31246 L .51077 .3124 L .51077 .31233 L .51077 .31227 L .51077 .3122 L .51077 .31214 L .51077 .31207 L .51076 .31201 L .51075 .31188 L .51074 .31176 L .51071 .31151 L .51066 .31127 L .51053 .31079 L .51036 .31034 L .51015 .30991 L Mfstroke P P p p .004 w .67485 .16503 m .66488 .16401 L .65503 .16319 L .65015 .16286 L .64529 .16257 L .64047 .16234 L .63568 .16215 L .6333 .16208 L .63093 .16201 L .62856 .16196 L .62621 .16192 L .62386 .16189 L .62152 .16187 L .61919 .16187 L .61687 .16187 L .61455 .16189 L .61225 .16191 L .60996 .16195 L .60767 .162 L .60314 .16213 L .59864 .1623 L .58976 .16277 L .58104 .16341 L .56413 .16513 L .54795 .16743 L .53251 .17026 L .50401 .17733 L .47878 .18598 L .45691 .19588 L .43839 .20668 L .42318 .21809 L .41117 .22982 L .40631 .23573 L .40219 .24163 L .39879 .24749 L .39607 .25329 L .394 .259 L .39321 .26181 L .39256 .2646 L .39207 .26735 L .39188 .26871 L .39172 .27007 L .3916 .27141 L .39155 .27208 L .39151 .27275 L .39148 .27341 L .39146 .27407 L .39145 .27473 L .39144 .27539 L .39145 .27605 L Mistroke .39146 .2767 L .39148 .27735 L .3915 .27799 L .39158 .27928 L .39169 .28055 L .39184 .28181 L .39201 .28306 L .39244 .28553 L .39299 .28795 L .39366 .29032 L .3953 .29491 L .39734 .29929 L .39974 .30345 L .40549 .31109 L .4123 .31779 L .41992 .32354 L .42811 .32835 L .43666 .33224 L .44536 .33525 L .44971 .33644 L .45404 .33742 L .45832 .33821 L .46255 .33882 L .46463 .33905 L .46669 .33925 L .46872 .3394 L .46973 .33946 L .47074 .33951 L .47173 .33955 L .47223 .33956 L .47272 .33957 L .47321 .33959 L .4737 .33959 L .47419 .3396 L .47468 .33961 L .47516 .33961 L .47565 .33961 L .47613 .3396 L .47661 .3396 L .47708 .33959 L .47756 .33958 L .4785 .33955 L .47944 .33952 L .48037 .33947 L .4822 .33936 L .48399 .33921 L .48575 .33904 L .48915 .33859 L .49241 .33803 L .49842 .33661 L Mistroke .50376 .33485 L .5084 .33282 L .51234 .3306 L .51559 .32824 L .51817 .32581 L .51922 .32458 L .52012 .32335 L .52087 .32213 L .52147 .32092 L .52193 .31973 L .52212 .31914 L .52227 .31856 L .52239 .31798 L .52243 .3177 L .52247 .31741 L .52251 .31713 L .52253 .31685 L .52254 .31671 L .52255 .31657 L .52256 .31644 L .52256 .3163 L .52257 .31616 L .52257 .31602 L .52257 .31589 L .52256 .31575 L .52256 .31562 L .52255 .31548 L .52255 .31535 L .52254 .31522 L .52252 .31495 L .52249 .31469 L .52241 .31417 L .52231 .31366 L .52218 .31316 L .52186 .3122 L .52146 .31128 L .52098 .3104 L .51982 .30879 L .51843 .30737 L .51687 .30615 L .51518 .30512 L .51341 .30429 L .5116 .30364 L Mfstroke P P p p .004 w .64578 .039 m .61457 .04237 L .58471 .04679 L .55626 .05218 L .52926 .05847 L .47979 .07337 L .43651 .09083 L .39947 .11025 L .36863 .13102 L .34382 .15262 L .32479 .17455 L .31734 .18549 L .31121 .19636 L .30634 .20711 L .30436 .21242 L .30268 .21768 L .30129 .22289 L .30018 .22804 L .29973 .2306 L .29934 .23314 L .29903 .23566 L .29878 .23817 L .29867 .23941 L .29859 .24065 L .29852 .24189 L .29847 .24312 L .29843 .24435 L .29841 .24558 L .2984 .24679 L .29841 .24801 L .29843 .24922 L .29847 .25042 L .29852 .25162 L .29859 .25282 L .29877 .25519 L .29901 .25754 L .29967 .26219 L .30054 .26674 L .30291 .27559 L .30607 .28405 L .30996 .29212 L .31967 .30703 L .33154 .32022 L .3451 .33165 L .35991 .34132 L .37555 .34927 L .39163 .35555 L .39972 .35808 L .40781 .36023 L .41583 .36201 L Mistroke .42377 .36343 L .43159 .3645 L .43545 .36491 L .43926 .36525 L .44303 .3655 L .4449 .3656 L .44675 .36568 L .44859 .36575 L .44951 .36577 L .45042 .36579 L .45133 .36581 L .45224 .36582 L .45314 .36582 L .45404 .36583 L .45494 .36582 L .45583 .36582 L .45672 .36581 L .4576 .36579 L .45936 .36575 L .46111 .36569 L .46284 .36561 L .46455 .36552 L .46793 .36529 L .47124 .365 L .47449 .36466 L .48077 .36379 L .48676 .36272 L .49246 .36146 L .50291 .35845 L .51208 .35488 L .51995 .35089 L .52653 .3466 L .53185 .34213 L .53596 .33758 L .53759 .3353 L .53893 .33303 L .54001 .33079 L .54084 .32858 L .54116 .32749 L .54142 .32642 L .54163 .32535 L .54171 .32482 L .54177 .3243 L .54183 .32378 L .54185 .32352 L .54187 .32326 L .54188 .323 L .54189 .32275 L .5419 .32249 L .54191 .32224 L Mistroke .54191 .32198 L .54191 .32173 L .5419 .32148 L .5419 .32123 L .54188 .32098 L .54187 .32073 L .54183 .32024 L .54178 .31975 L .54172 .31926 L .54157 .31831 L .54136 .31737 L .54112 .31645 L .5405 .31467 L .53973 .31297 L .53778 .30982 L .53537 .30703 L .53259 .30461 L .52954 .30255 L .52631 .30085 L .52297 .2995 L .52129 .29895 L .51961 .29848 L .51794 .29809 L .51628 .29778 L .51465 .29754 L .51305 .29736 L Mfstroke P P p p .004 w .55813 .56107 m .60195 .53974 L .63875 .51669 L .66868 .49254 L .692 .46787 L .70129 .4555 L .70907 .44318 L .71538 .43098 L .72029 .41894 L .72224 .41299 L .72387 .4071 L .72517 .40128 L .72616 .39552 L .72654 .39267 L .72685 .38984 L .72698 .38843 L .72709 .38703 L .72718 .38563 L .72725 .38424 L .7273 .38285 L .72734 .38147 L .72736 .38009 L .72736 .37872 L .72735 .37735 L .72731 .37599 L .72726 .37463 L .7272 .37328 L .72701 .37059 L .72676 .36793 L .72607 .36268 L .72513 .35752 L .72394 .35247 L .72089 .34267 L .71697 .33331 L .70684 .31595 L .6941 .30051 L .67931 .28704 L .66296 .27555 L .64555 .26603 L .62752 .25841 L .60927 .25262 L .60018 .25037 L .59116 .24854 L .58226 .2471 L .57786 .24653 L .57351 .24605 L .5692 .24566 L .56494 .24535 L .56283 .24523 L .56073 .24514 L Mistroke .55865 .24506 L .55762 .24503 L .55658 .245 L .55556 .24498 L .55453 .24497 L .55351 .24496 L .55249 .24495 L .55148 .24495 L .55047 .24495 L .54947 .24496 L .54847 .24498 L .54747 .245 L .54648 .24502 L .5445 .24508 L .54255 .24516 L .54061 .24525 L .53679 .2455 L .53304 .24582 L .52577 .24665 L .51882 .24772 L .5122 .24902 L .50592 .25052 L .50292 .25134 L .50145 .25177 L .50072 .25199 L Mfstroke .49928 .25243 m .49858 .25266 L .49717 .25312 L .49444 .25406 L .48923 .25607 L .47993 .26044 L .47209 .26519 L .46568 .27019 L .46065 .27531 L .45864 .27788 L .45694 .28045 L .45555 .28299 L .45445 .28551 L .45401 .28675 L .45363 .28798 L .45333 .2892 L .45309 .2904 L .45299 .291 L .45291 .29159 L .45285 .29218 L .45282 .29247 L .4528 .29276 L .45278 .29305 L .45276 .29334 L .45275 .29363 L .45274 .29392 L .45274 .29421 L .45274 .29449 L .45274 .29478 L .45275 .29506 L .45276 .29534 L .45277 .29562 L .45281 .29618 L .45286 .29674 L .45292 .29728 L .45309 .29837 L .45331 .29943 L .4539 .30149 L .45466 .30346 L .4556 .30534 L .45669 .30712 L .45927 .31038 L .4623 .31324 L .46566 .31568 L .46925 .31771 L .47299 .31935 L .47678 .3206 L .47867 .32109 L .48055 .3215 L .48241 .32182 L Mistroke .48333 .32195 L .48424 .32206 L .48514 .32215 L .48603 .32222 L .48647 .32225 L .48691 .32227 L .48735 .3223 L .48778 .32231 L .488 .32232 L .48821 .32232 L .48842 .32233 L .48864 .32233 L .48885 .32233 L .48906 .32233 L .48927 .32234 L .48948 .32233 L .48969 .32233 L .4899 .32233 L .49011 .32233 L .49031 .32232 L .49072 .32231 L .49113 .32229 L .49193 .32225 L .49272 .32219 L .49425 .32203 L .49571 .32182 L .49711 .32157 L Mfstroke P P p p .004 w .52907 .43504 m .55098 .42438 L .56937 .41285 L .58434 .40078 L .596 .38844 L .60065 .38226 L .60453 .3761 L .60769 .37 L .61015 .36398 L .61112 .361 L .61193 .35806 L .61258 .35515 L .61308 .35227 L .61327 .35085 L .61343 .34943 L .61349 .34873 L .61354 .34802 L .61359 .34732 L .61362 .34663 L .61365 .34593 L .61367 .34524 L .61368 .34455 L .61368 .34387 L .61367 .34318 L .61366 .3425 L .61363 .34182 L .6136 .34115 L .61351 .33981 L .61338 .33847 L .61304 .33585 L .61256 .33327 L .61197 .33074 L .61044 .32584 L .60848 .32116 L .60342 .31248 L .59705 .30476 L .58965 .29803 L .58148 .29228 L .57278 .28752 L .56376 .28371 L .55464 .28082 L .55009 .27969 L .54558 .27878 L .54113 .27806 L .53893 .27777 L .53675 .27753 L .5346 .27734 L .53247 .27719 L .53142 .27713 L .53037 .27708 L Mistroke .52933 .27704 L .52881 .27702 L .52829 .27701 L .52778 .277 L .52727 .27699 L .52676 .27699 L .52625 .27698 L .52574 .27698 L .52524 .27699 L .52473 .27699 L .52423 .277 L .52373 .27701 L .52324 .27702 L .52225 .27705 L .52127 .27709 L .52031 .27714 L .51839 .27726 L .51652 .27742 L .51288 .27783 L .50941 .27837 L .5061 .27902 L .50296 .27977 L .50146 .28018 L .50072 .28039 L .50036 .2805 L Mfstroke .49964 .28072 m .49929 .28084 L .49859 .28107 L .49722 .28154 L .49462 .28254 L .48997 .28473 L .48604 .28711 L .48284 .2896 L .48033 .29216 L .47932 .29345 L .47847 .29473 L .47777 .29601 L .47722 .29726 L .477 .29788 L .47682 .2985 L .47666 .29911 L .47654 .29971 L .4765 .30001 L .47646 .3003 L .47642 .3006 L .47641 .30074 L .4764 .30089 L .47639 .30104 L .47638 .30118 L .47637 .30133 L .47637 .30147 L .47637 .30161 L .47637 .30175 L .47637 .3019 L .47637 .30204 L .47638 .30218 L .47638 .30232 L .4764 .3026 L .47643 .30288 L .47646 .30315 L .47655 .30369 L .47665 .30422 L .47695 .30525 L .47733 .30624 L .4778 .30718 L .47834 .30807 L .47964 .3097 L .48115 .31113 L .48283 .31235 L .48463 .31337 L .48649 .31418 L .48839 .31481 L .48934 .31505 L .49028 .31526 L .49121 .31542 L Mistroke .49166 .31548 L .49212 .31554 L .49257 .31558 L .49302 .31562 L .49324 .31563 L .49346 .31565 L .49367 .31566 L .49389 .31566 L .494 .31567 L .49411 .31567 L .49421 .31567 L .49432 .31567 L .49443 .31568 L .49453 .31568 L .49464 .31568 L .49474 .31568 L .49485 .31567 L .49495 .31567 L .49505 .31567 L .49516 .31567 L .49536 .31566 L .49557 .31566 L .49597 .31563 L .49636 .3156 L .49712 .31553 L .49786 .31542 L .49855 .31529 L Mfstroke P P p p P P p p .004 w .47093 .18299 m .44902 .19366 L .43063 .20518 L .41566 .21725 L .404 .22959 L .39935 .23578 L .39547 .24193 L .39231 .24804 L .38985 .25406 L .38888 .25703 L .38807 .25997 L .38742 .26289 L .38692 .26576 L .38673 .26719 L .38657 .2686 L .38651 .26931 L .38646 .27001 L .38641 .27071 L .38638 .27141 L .38635 .2721 L .38633 .27279 L .38632 .27348 L .38632 .27417 L .38633 .27485 L .38634 .27553 L .38637 .27621 L .3864 .27689 L .38649 .27823 L .38662 .27956 L .38696 .28219 L .38744 .28476 L .38803 .28729 L .38956 .29219 L .39152 .29687 L .39658 .30555 L .40295 .31327 L .41035 .32001 L .41852 .32575 L .42722 .33051 L .43624 .33432 L .44536 .33722 L .44991 .33834 L .45442 .33926 L .45887 .33997 L .46107 .34026 L .46325 .3405 L .4654 .3407 L .46753 .34085 L .46858 .34091 L .46963 .34096 L Mistroke .47067 .341 L .47119 .34101 L .47171 .34102 L .47222 .34103 L .47273 .34104 L .47324 .34105 L .47375 .34105 L .47426 .34105 L .47476 .34105 L .47527 .34104 L .47577 .34104 L .47627 .34103 L .47676 .34102 L .47775 .34099 L .47873 .34095 L .47969 .3409 L .48161 .34077 L .48348 .34062 L .48712 .3402 L .49059 .33967 L .4939 .33902 L .49704 .33827 L .49854 .33785 L .49928 .33764 L .49964 .33753 L Mfstroke .50036 .33731 m .50071 .3372 L .50141 .33697 L .50278 .33649 L .50538 .33549 L .51003 .3333 L .51396 .33093 L .51716 .32843 L .51967 .32587 L .52068 .32458 L .52153 .3233 L .52223 .32203 L .52278 .32077 L .523 .32015 L .52318 .31954 L .52334 .31893 L .52346 .31833 L .5235 .31803 L .52354 .31773 L .52358 .31744 L .52359 .31729 L .5236 .31714 L .52361 .317 L .52362 .31685 L .52363 .31671 L .52363 .31657 L .52363 .31642 L .52363 .31628 L .52363 .31614 L .52363 .316 L .52362 .31585 L .52362 .31571 L .5236 .31543 L .52357 .31516 L .52354 .31488 L .52345 .31434 L .52335 .31381 L .52305 .31278 L .52267 .3118 L .5222 .31086 L .52166 .30997 L .52036 .30833 L .51885 .30691 L .51717 .30569 L .51537 .30467 L .51351 .30385 L .51161 .30322 L .51066 .30298 L .50972 .30278 L .50879 .30262 L Mistroke .50834 .30255 L .50788 .3025 L .50743 .30245 L .50698 .30242 L .50676 .3024 L .50654 .30239 L .50633 .30238 L .50611 .30237 L .506 .30237 L .50589 .30236 L .50579 .30236 L .50568 .30236 L .50557 .30236 L .50547 .30236 L .50536 .30236 L .50526 .30236 L .50515 .30236 L .50505 .30236 L .50495 .30236 L .50484 .30236 L .50464 .30237 L .50443 .30238 L .50403 .3024 L .50364 .30243 L .50288 .30251 L .50214 .30261 L .50145 .30274 L Mfstroke P P p p .004 w .44187 .05697 m .39805 .07829 L .36125 .10134 L .33132 .12549 L .308 .15017 L .29871 .16254 L .29093 .17485 L .28462 .18706 L .27971 .1991 L .27776 .20504 L .27613 .21093 L .27483 .21675 L .27384 .22251 L .27346 .22536 L .27315 .22819 L .27302 .2296 L .27291 .231 L .27282 .2324 L .27275 .2338 L .2727 .23518 L .27266 .23657 L .27264 .23795 L .27264 .23932 L .27265 .24069 L .27269 .24205 L .27274 .2434 L .2728 .24476 L .27299 .24744 L .27324 .2501 L .27393 .25535 L .27487 .26051 L .27606 .26557 L .27911 .27536 L .28303 .28473 L .29316 .30208 L .3059 .31752 L .32069 .331 L .33704 .34248 L .35445 .35201 L .37248 .35962 L .39073 .36542 L .39982 .36766 L .40884 .3695 L .41774 .37093 L .42214 .37151 L .42649 .37199 L .4308 .37238 L .43506 .37268 L .43717 .3728 L .43927 .3729 L Mistroke .44135 .37297 L .44238 .37301 L .44342 .37303 L .44444 .37305 L .44547 .37307 L .44649 .37308 L .44751 .37308 L .44852 .37308 L .44953 .37308 L .45053 .37307 L .45153 .37306 L .45253 .37304 L .45352 .37302 L .4555 .37295 L .45745 .37288 L .45939 .37278 L .46321 .37253 L .46696 .37222 L .47423 .37139 L .48118 .37032 L .4878 .36902 L .49408 .36751 L .49708 .36669 L .49855 .36626 L .49928 .36605 L Mfstroke .50072 .3656 m .50142 .36538 L .50283 .36492 L .50556 .36397 L .51077 .36197 L .52007 .35759 L .52791 .35284 L .53432 .34784 L .53935 .34272 L .54136 .34015 L .54306 .33758 L .54445 .33504 L .54555 .33253 L .54599 .33129 L .54637 .33006 L .54667 .32884 L .54691 .32763 L .54701 .32704 L .54709 .32644 L .54715 .32586 L .54718 .32556 L .5472 .32527 L .54722 .32498 L .54724 .32469 L .54725 .3244 L .54726 .32411 L .54726 .32383 L .54726 .32354 L .54726 .32326 L .54725 .32297 L .54724 .32269 L .54723 .32241 L .54719 .32185 L .54714 .3213 L .54708 .32075 L .54691 .31967 L .54669 .31861 L .5461 .31655 L .54534 .31457 L .5444 .3127 L .54331 .31092 L .54073 .30765 L .5377 .3048 L .53434 .30235 L .53075 .30032 L .52701 .29869 L .52322 .29743 L .52133 .29694 L .51945 .29654 L .51759 .29622 L Mistroke .51667 .29609 L .51576 .29598 L .51486 .29589 L .51397 .29581 L .51353 .29578 L .51309 .29576 L .51265 .29574 L .51222 .29572 L .512 .29572 L .51179 .29571 L .51158 .29571 L .51136 .2957 L .51115 .2957 L .51094 .2957 L .51073 .2957 L .51052 .2957 L .51031 .2957 L .5101 .2957 L .50989 .29571 L .50969 .29571 L .50928 .29572 L .50887 .29574 L .50807 .29578 L .50728 .29584 L .50575 .296 L .50429 .29621 L .50289 .29647 L Mfstroke P P p p .004 w .35422 .57903 m .38543 .57567 L .41529 .57124 L .44374 .56585 L .47074 .55957 L .52021 .54467 L .56349 .5272 L .60053 .50779 L .63137 .48701 L .65618 .46541 L .67521 .44349 L .68266 .43254 L .68879 .42167 L .69366 .41093 L .69564 .40562 L .69732 .40035 L .69871 .39514 L .69982 .38999 L .70027 .38743 L .70066 .3849 L .70097 .38237 L .70122 .37987 L .70133 .37862 L .70141 .37738 L .70148 .37614 L .70153 .37491 L .70157 .37368 L .70159 .37246 L .7016 .37124 L .70159 .37003 L .70157 .36882 L .70153 .36761 L .70148 .36641 L .70141 .36522 L .70123 .36284 L .70099 .36049 L .70033 .35585 L .69946 .35129 L .69709 .34245 L .69393 .33398 L .69004 .32591 L .68033 .31101 L .66846 .29782 L .6549 .28638 L .64009 .27671 L .62445 .26876 L .60837 .26249 L .60028 .25995 L .59219 .2578 L .58417 .25603 L Mistroke .57623 .25461 L .56841 .25353 L .56455 .25312 L .56074 .25279 L .55697 .25253 L .5551 .25243 L .55325 .25235 L .55141 .25229 L .55049 .25226 L .54958 .25224 L .54867 .25223 L .54776 .25222 L .54686 .25221 L .54596 .25221 L .54506 .25221 L .54417 .25222 L .54328 .25223 L .5424 .25224 L .54064 .25229 L .53889 .25234 L .53716 .25242 L .53545 .25251 L .53207 .25274 L .52876 .25303 L .52551 .25338 L .51923 .25424 L .51324 .25531 L .50754 .25657 L .49709 .25959 L .48792 .26315 L .48005 .26714 L .47347 .27143 L .46815 .2759 L .46404 .28046 L .46241 .28274 L .46107 .285 L .45999 .28724 L .45916 .28945 L .45884 .29054 L .45858 .29162 L .45837 .29268 L .45829 .29321 L .45823 .29374 L .45817 .29426 L .45815 .29452 L .45813 .29477 L .45812 .29503 L .45811 .29529 L .4581 .29554 L .45809 .2958 L Mistroke .45809 .29605 L .45809 .2963 L .4581 .29655 L .4581 .29681 L .45812 .29705 L .45813 .2973 L .45817 .2978 L .45822 .29829 L .45828 .29877 L .45843 .29973 L .45864 .30067 L .45888 .30159 L .4595 .30337 L .46027 .30507 L .46222 .30821 L .46463 .311 L .46741 .31343 L .47046 .31549 L .47369 .31719 L .47703 .31854 L .47871 .31908 L .48039 .31955 L .48206 .31994 L .48372 .32026 L .48535 .3205 L .48695 .32067 L Mfstroke P P p p .004 w .32515 .45301 m .33512 .45403 L .34497 .45485 L .34985 .45518 L .35471 .45546 L .35953 .4557 L .36432 .45588 L .3667 .45596 L .36907 .45602 L .37144 .45607 L .37379 .45611 L .37614 .45614 L .37848 .45616 L .38081 .45617 L .38313 .45616 L .38545 .45615 L .38775 .45612 L .39004 .45608 L .39233 .45603 L .39686 .4559 L .40136 .45573 L .41024 .45526 L .41896 .45463 L .43587 .45291 L .45205 .45061 L .46749 .44778 L .49599 .4407 L .52122 .43205 L .54309 .42216 L .56161 .41136 L .57682 .39995 L .58883 .38821 L .59369 .3823 L .59781 .3764 L .60121 .37054 L .60393 .36474 L .606 .35904 L .60679 .35622 L .60744 .35344 L .60793 .35068 L .60812 .34932 L .60828 .34797 L .6084 .34662 L .60845 .34595 L .60849 .34529 L .60852 .34462 L .60854 .34396 L .60855 .3433 L .60856 .34264 L .60855 .34199 L Mistroke .60854 .34134 L .60852 .34069 L .6085 .34004 L .60842 .33876 L .60831 .33748 L .60816 .33622 L .60799 .33497 L .60756 .3325 L .60701 .33008 L .60634 .32771 L .6047 .32312 L .60266 .31874 L .60026 .31458 L .59451 .30695 L .5877 .30025 L .58008 .2945 L .57189 .28969 L .56334 .28579 L .55464 .28279 L .55029 .2816 L .54596 .28061 L .54168 .27982 L .53745 .27922 L .53537 .27898 L .53331 .27879 L .53128 .27864 L .53027 .27858 L .52926 .27853 L .52827 .27849 L .52777 .27847 L .52728 .27846 L .52679 .27845 L .5263 .27844 L .52581 .27843 L .52532 .27843 L .52484 .27843 L .52435 .27843 L .52387 .27843 L .52339 .27844 L .52292 .27844 L .52244 .27845 L .5215 .27848 L .52056 .27852 L .51963 .27856 L .5178 .27867 L .51601 .27882 L .51425 .279 L .51085 .27945 L .50759 .28001 L .50158 .28143 L Mistroke .49624 .28319 L .4916 .28521 L .48766 .28744 L .48441 .28979 L .48183 .29223 L .48078 .29346 L .47988 .29468 L .47913 .29591 L .47853 .29711 L .47807 .29831 L .47788 .29889 L .47773 .29948 L .47761 .30005 L .47757 .30034 L .47753 .30062 L .47749 .3009 L .47747 .30118 L .47746 .30132 L .47745 .30146 L .47744 .3016 L .47744 .30174 L .47743 .30187 L .47743 .30201 L .47743 .30215 L .47744 .30228 L .47744 .30242 L .47745 .30255 L .47745 .30268 L .47746 .30282 L .47748 .30308 L .47751 .30335 L .47759 .30386 L .47769 .30437 L .47782 .30487 L .47814 .30584 L .47854 .30676 L .47902 .30763 L .48018 .30925 L .48157 .31067 L .48313 .31189 L .48482 .31291 L .48659 .31375 L .4884 .3144 L Mfstroke P P p p .004 w .29609 .32698 m .31334 .34052 L .33199 .35189 L .35152 .36114 L .37149 .36835 L .3815 .37122 L .39146 .37362 L .40133 .37557 L .41107 .37709 L .41588 .37769 L .42065 .3782 L .42536 .3786 L .43001 .3789 L .43232 .37902 L .43461 .37912 L .43575 .37916 L .43689 .37919 L .43802 .37922 L .43915 .37924 L .44027 .37926 L .44139 .37927 L .4425 .37927 L .44361 .37928 L .44472 .37927 L .44582 .37926 L .44692 .37924 L .44801 .37922 L .45018 .37917 L .45234 .37909 L .45659 .37888 L .46076 .37859 L .46485 .37822 L .47277 .37727 L .48035 .37606 L .49439 .37293 L .50688 .369 L .51778 .36443 L .52707 .35938 L .53478 .354 L .54094 .34842 L .54563 .34278 L .54745 .33997 L .54893 .33719 L .55009 .33444 L .55055 .33308 L .55094 .33173 L .55126 .3304 L .5515 .32908 L .5516 .32843 L .55168 .32778 L Mistroke .55174 .32714 L .55177 .32682 L .55179 .3265 L .5518 .32618 L .55182 .32587 L .55183 .32555 L .55183 .32524 L .55183 .32493 L .55183 .32461 L .55182 .3243 L .55181 .32399 L .55179 .32369 L .55178 .32338 L .55173 .32277 L .55166 .32217 L .55158 .32157 L .55138 .32038 L .55113 .31923 L .55046 .31698 L .54959 .31483 L .54854 .31279 L .54733 .31085 L .54666 .30992 L .54631 .30946 L .54614 .30924 L Mfstroke .54578 .3088 m .5456 .30858 L .54522 .30814 L .54446 .3073 L .54284 .30569 L .5393 .30282 L .53545 .30039 L .53141 .29841 L .52727 .29686 L .52519 .29624 L .52311 .29571 L .52105 .29528 L .51902 .29494 L .51802 .2948 L .51702 .29469 L .51604 .29459 L .51506 .29452 L .51458 .29449 L .5141 .29447 L .51362 .29445 L .51339 .29444 L .51315 .29443 L .51291 .29442 L .51268 .29442 L .51245 .29442 L .51221 .29441 L .51198 .29441 L .51175 .29441 L .51152 .29441 L .51129 .29441 L .51106 .29442 L .51084 .29442 L .51038 .29443 L .50994 .29445 L .50949 .29447 L .50862 .29452 L .50776 .29459 L .50609 .29477 L .5045 .29501 L .50298 .2953 L .50154 .29563 L .4989 .29643 L .49659 .29737 L .49461 .2984 L .49296 .29952 L .49164 .30067 L .49063 .30184 L .49023 .30243 L .4899 .30301 L .48965 .30358 L Mistroke .48954 .30387 L .48945 .30415 L .48938 .30443 L .48932 .3047 L .48929 .30484 L .48927 .30497 L .48926 .30511 L .48924 .30524 L .48924 .30531 L .48923 .30537 L .48923 .30544 L .48923 .30551 L .48923 .30557 L .48923 .30564 L .48923 .3057 L .48923 .30577 L .48923 .30583 L .48923 .3059 L .48923 .30596 L .48924 .30602 L .48925 .30615 L .48926 .30628 L .48929 .30652 L .48934 .30677 L .48947 .30724 L .48964 .30769 L .48985 .30812 L Mfstroke P P p p .004 w .26702 .20096 m .26537 .20711 L .26404 .2132 L .26351 .21622 L .26305 .21922 L .26267 .2222 L .26237 .22516 L .26224 .22663 L .26214 .2281 L .26206 .22956 L .26199 .23102 L .26194 .23247 L .26191 .23391 L .2619 .23535 L .26191 .23679 L .26194 .23822 L .26198 .23964 L .26204 .24106 L .26212 .24247 L .26233 .24528 L .26261 .24806 L .26337 .25354 L .2644 .25893 L .26718 .26938 L .2709 .27938 L .27548 .28892 L .28692 .30654 L .30093 .32213 L .31693 .33565 L .33441 .34709 L .35287 .35649 L .37186 .36392 L .38143 .36692 L .39097 .36946 L .40045 .37157 L .40983 .37325 L .41907 .37453 L .42362 .37502 L .42813 .37542 L .43258 .37572 L .43479 .37584 L .43698 .37594 L .43916 .37601 L .44024 .37604 L .44132 .37606 L .44239 .37608 L .44346 .3761 L .44453 .37611 L .44559 .37611 L .44665 .37611 L Mistroke .44771 .3761 L .44876 .37609 L .4498 .37607 L .45188 .37602 L .45394 .37595 L .45598 .37587 L .45801 .37576 L .462 .37549 L .46592 .37515 L .46975 .37474 L .47718 .37373 L .48426 .37247 L .49099 .37098 L .50335 .36743 L .51419 .36322 L .5235 .35851 L .53128 .35345 L .53757 .34817 L .54244 .34279 L .54436 .3401 L .54596 .33742 L .54724 .33477 L .54822 .33217 L .5486 .33088 L .54891 .32961 L .54916 .32835 L .54925 .32772 L .54933 .3271 L .5494 .32649 L .54942 .32618 L .54944 .32588 L .54946 .32557 L .54948 .32527 L .54949 .32497 L .54949 .32467 L .5495 .32437 L .54949 .32407 L .54949 .32377 L .54948 .32348 L .54947 .32318 L .54945 .32289 L .54941 .3223 L .54935 .32173 L .54928 .32115 L .5491 .32002 L .54886 .31891 L .54857 .31782 L .54785 .31572 L .54694 .31371 L .54464 .30999 L Mistroke .5418 .3067 L .53852 .30383 L .53492 .30139 L .5311 .29938 L .52716 .29779 L .52518 .29714 L .52319 .29658 L .52122 .29612 L .51926 .29575 L .51733 .29546 L .51638 .29535 L .51544 .29526 L .51451 .29518 L .51404 .29515 L .51358 .29513 L .51313 .29511 L .51268 .29509 L .51245 .29508 L .51223 .29508 L .512 .29508 L .51178 .29507 L .51156 .29507 L .51134 .29507 L .51112 .29507 L .5109 .29507 L .51068 .29507 L .51046 .29508 L .51025 .29508 L .51003 .29509 L .5096 .2951 L .50918 .29512 L .50834 .29517 L .50752 .29523 L .50671 .29531 L .50515 .2955 L .50366 .29575 L .50224 .29605 L .49963 .29677 L .49733 .29763 L .49535 .29859 L .49369 .29964 L .49235 .30073 L .49129 .30185 L Mfstroke P P p p .004 w .23795 .07493 m .22358 .09238 L .21139 .1098 L .2013 .1271 L .19324 .14422 L .18712 .16108 L .18475 .16939 L .18284 .17762 L .18137 .18575 L .1808 .18977 L .18033 .19377 L .17997 .19775 L .17982 .19972 L .1797 .20169 L .17961 .20366 L .17954 .20561 L .1795 .20756 L .17948 .2095 L .17949 .21143 L .17952 .21336 L .17957 .21527 L .17965 .21718 L .17975 .21908 L .17988 .22098 L .1802 .22474 L .18062 .22847 L .18112 .23217 L .18239 .23946 L .18401 .24661 L .18595 .25361 L .19078 .26717 L .19678 .28011 L .20385 .2924 L .22082 .31496 L .2409 .33477 L .26338 .35177 L .28755 .366 L .31278 .37751 L .33847 .38641 L .35133 .38992 L .36411 .39284 L .37676 .39519 L .38922 .39698 L .39537 .39768 L .40146 .39826 L .40748 .3987 L .41046 .39888 L .41342 .39903 L .41636 .39915 L .41782 .3992 L Mistroke .41928 .39924 L .42074 .39927 L .42218 .3993 L .42363 .39932 L .42506 .39933 L .42649 .39933 L .42792 .39933 L .42934 .39932 L .43076 .39931 L .43216 .39929 L .43357 .39926 L .43636 .39918 L .43912 .39908 L .44186 .39895 L .44727 .39862 L .45257 .39819 L .45777 .39767 L .46783 .39635 L .47743 .3947 L .4952 .39052 L .51097 .38533 L .52469 .37935 L .53634 .37279 L .54596 .36583 L .5536 .35864 L .55671 .35501 L .55937 .35139 L .56158 .34779 L .56336 .34422 L .56474 .34069 L .56528 .33896 L .56573 .33724 L .56609 .33554 L .56623 .33469 L .56635 .33386 L .56645 .33302 L .56652 .3322 L .56656 .33179 L .56658 .33138 L .5666 .33097 L .56661 .33056 L .56662 .33016 L .56663 .32975 L .56663 .32935 L .56662 .32895 L .56661 .32855 L .5666 .32816 L .56658 .32776 L .56655 .32737 L .56648 .32658 L Mistroke .5664 .32581 L .56617 .32428 L .56588 .32278 L .56551 .3213 L .56457 .31845 L .56339 .31573 L .56197 .31314 L .55855 .30837 L .55445 .30418 L .54983 .30056 L .54485 .29753 L .53962 .29507 L .53429 .29314 L .53161 .29238 L .52895 .29174 L .52631 .29122 L .52371 .29082 L .52243 .29066 L .52115 .29053 L .5199 .29042 L .51927 .29038 L .51865 .29034 L .51804 .29031 L .51742 .29028 L .51712 .29027 L .51682 .29026 L .51651 .29026 L .51621 .29025 L .51591 .29025 L .51561 .29024 L .51532 .29024 L 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This shows that that a center can be perturbed to either a stable or unstable spiral. ;[s] 3:0,0;46,1;51,0;152,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = text; inactive; preserveAspect] b) The command Solve[] would give us various complex-valued critical points which depend on the unspecified quantity h, so we'll solve for the real-valued critical points `by hand'. Let ;[s] 6:0,0;2,1;16,2;23,1;118,2;119,1;187,-1; 3:1,13,9,Times,2,12,0,0,0;3,13,9,Times,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect] F[x_,y_] = y + h*x(x^2+y^2); G[x_,y_] = -x + h*y(x^2+y^2); :[font = text; inactive; preserveAspect] Then F[x,y]==0 and G[x,y]==0 imply y == -h*x(x^2+y^2) x == h*y(x^2+y^2) Substituting one into the other gives x == -x*h^2(x^2+y^2) y == -y*h^2(x^2+y^2) These equations would imply that a negative number equals 1, unless both x and y are both zero. Thus, {0,0} is the only critical point. The corresponding linear system at {0,0} clearly has coefficient matrix {{0,1},{-1,0}}, which represents a center (see part a): ;[s] 23:0,0;5,1;14,0;19,1;28,0;35,1;87,0;126,1;176,0;235,1;236,0;250,1;251,0;256,1;257,0;280,1;285,0;349,1;354,0;386,1;400,0;438,2;439,0;442,-1; 3:12,13,9,Times,0,12,0,0,0;10,13,10,Courier,1,12,0,0,0;1,13,9,Times,2,12,0,0,0; :[font = input; preserveAspect; startGroup] A = {{D[F[x,y],x], D[F[x,y],y]}, {D[G[x,y],x], D[G[x,y],y]}}; A//MatrixForm :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{2*h*x^2 + h*(x^2 + y^2), 1 + 2*h*x*y}, {-1 + 2*h*x*y, 2*h*y^2 + h*(x^2 + y^2)}}] ;[o] 2 2 2 2 h x + h (x + y ) 1 + 2 h x y 2 2 2 -1 + 2 h x y 2 h y + h (x + y ) :[font = input; preserveAspect; startGroup] A/.{x->0,y->0}//MatrixForm :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{0, 1}, {-1, 0}}] ;[o] 0 1 -1 0 :[font = text; inactive; preserveAspect] The case h = -1: ;[s] 3:0,1;9,2;15,1;17,-1; 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This shows that the stability of a critical point can change from stable to unstable even though the linear approximation remains a center. ;[s] 3:0,0;51,1;56,0;211,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Problem 8 :[font = text; inactive; preserveAspect; endGroup] Consider the competing species model: x'[t] == x[t]((5/4) - (3/4)x[t] - (1/4)y[t]) y'[t] == y[t]((7/4) - (3/2)x[t] - (1/2)y[t]) a) Find all critical points of this system. At each critical point, calculate the corresponding linear system and find the eigenvalues of the coefficient matrix. Then identify the type and stability of the critical point. b) Plot the vector field using PlotVectorField[] with the options ScaleFunction->(Log[50#+1]&) and Frame->True . Choose a representative sample of several initial points {x0,y0} in the first quadrant. Use NDSolve[] to obtain numerical solutions through these points and plot the corresponding trajectories on a single graph. Indicate what happens to the trajectories for increasing t. Explain why realistically there can be no "peaceful coexistence". ;[s] 18:0,0;39,1;138,2;140,0;361,2;363,0;393,1;410,0;427,1;456,0;461,1;472,0;532,1;539,0;567,1;576,0;744,1;745,0;813,-1; 3:9,13,9,Times,0,12,0,0,0;7,13,10,Courier,1,12,0,0,0;2,13,9,Times,2,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Solution :[font = input; preserveAspect] Clear[x,y] :[font = input; preserveAspect] F[x_,y_] = x((5/4) - (3/4)x - (1/4)y); G[x_,y_] = y((7/4) - (3/2)x - (1/4)y); :[font = text; inactive; preserveAspect] a) The critical points are: ;[s] 2:0,0;2,1;29,-1; 2:1,13,9,Times,2,12,0,0,0;1,13,9,Times,0,12,0,0,0; :[font = input; preserveAspect; startGroup] cp = Solve[{F[x,y]==0, G[x,y]==0}] :[font = output; output; inactive; preserveAspect; endGroup] {{x -> 0, y -> 0}, {x -> 2/3, y -> 3}, {x -> 5/3, y -> 0}, {y -> 7, x -> 0}} ;[o] 2 5 {{x -> 0, y -> 0}, {x -> -, y -> 3}, {x -> -, y -> 0}, {y -> 7, x -> 0}} 3 3 :[font = input; preserveAspect; startGroup] A = {{D[F[x,y],x], D[F[x,y],y]}, {D[G[x,y],x], D[G[x,y],y]}}; A//MatrixForm :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{5/4 - (3*x)/2 - y/4, -x/4}, {(-3*y)/2, 7/4 - (3*x)/2 - y/2}}] ;[o] 5 3 x y -x - - --- - - -- 4 2 4 4 -3 y 7 3 x y ---- - - --- - - 2 4 2 2 :[font = input; preserveAspect; startGroup] A1 = A/.cp[[1]]; A1//MatrixForm :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{5/4, 0}, {0, 7/4}}] ;[o] 5 - 4 0 7 - 0 4 :[font = text; inactive; preserveAspect] The eigenvalues are 5/4 and 7/4, so the critical point {0,0} is an unstable improper node. ;[s] 7:0,0;20,1;23,0;28,1;31,0;55,1;60,0;91,-1; 2:4,13,9,Times,0,12,0,0,0;3,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] A2 = A/.cp[[2]]; A2//MatrixForm :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{-1/2, -1/6}, {-9/2, -3/4}}] ;[o] 1 1 -(-) -(-) 2 6 9 3 -(-) -(-) 2 4 :[font = input; preserveAspect; startGroup] Eigenvalues[A2] :[font = output; output; inactive; preserveAspect; endGroup] {-3/2, 1/4} ;[o] 3 1 {-(-), -} 2 4 :[font = text; inactive; preserveAspect] The eigenvalues are -3/2 and 1/4, so the critical point {2/3,3} is an unstable saddle point. ;[s] 7:0,0;20,1;24,0;29,1;32,0;56,1;63,0;93,-1; 2:4,13,9,Times,0,12,0,0,0;3,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] A3 = A/.cp[[3]]; A3//MatrixForm :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{-5/4, -5/12}, {0, -3/4}}] ;[o] 5 5 -(-) -(--) 4 12 3 -(-) 0 4 :[font = text; inactive; preserveAspect] The eigenvalues are -5/4 and -3/4, so the critical point {5/3,0} is an asymptotically stable improper node. ;[s] 7:0,0;20,1;24,0;29,1;33,0;57,1;64,0;108,-1; 2:4,13,9,Times,0,12,0,0,0;3,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] A4 = A/.cp[[4]]; A4//MatrixForm :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{-1/2, 0}, {-21/2, -7/4}}] ;[o] 1 -(-) 2 0 21 7 -(--) -(-) 2 4 :[font = text; inactive; preserveAspect] The eigenvalues are -1/2 and -7/4, so the critical point {0,7} is an asymptotically stable improper node. ;[s] 7:0,0;20,1;24,0;29,1;33,0;57,1;62,0;106,-1; 2:4,13,9,Times,0,12,0,0,0;3,13,10,Courier,1,12,0,0,0; :[font = text; inactive; preserveAspect] b) ;[s] 1:0,0;3,-1; 1:1,13,9,Times,2,12,0,0,0; :[font = input; preserveAspect; startGroup] v = PlotVectorField[{F[x,y],G[x,y]},{x,0,2},{y,0,7}, ScaleFunction->(Log[50#+1]&), Frame->True,AspectRatio->1] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 282] %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.47619 0.0238095 0.136054 [ [(0)] .02381 0 0 2 Msboxa [(0.5)] .2619 0 0 2 Msboxa [(1)] .5 0 0 2 Msboxa [(1.5)] .7381 0 0 2 Msboxa [(2)] .97619 0 0 2 Msboxa [(0)] 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endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = text; inactive; preserveAspect] We'll choose initial points on lines above, below and parallel to to y == -4x + 7. ;[s] 3:0,0;69,1;81,0;83,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect] solns1 = Table[{x[t],y[t]} /. First[NDSolve[ {x'[t] == x[t]((5/4) - (3/4)x[t] - (1/4)y[t]), y'[t] == y[t]((7/4) - (3/2)x[t] - (1/4)y[t]), x[0] == i, y[0] == -4i + 3}, {x[t],y[t]},{t,0,20}]], {i,0.25,0.75,0.125}]; :[font = input; preserveAspect] solns2 = Table[{x[t],y[t]} /. 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The only execptions to this would be the points on the`separatrix', the curve that forms the boundary bewteen the points that flow to {0,7} and those that flow to {5/3,0}. Since this is an ideal mathematical line and the model cannot be exact, points corresponding to competing populations cannot fall exactly on it. ;[s] 9:0,0;133,1;138,0;142,1;149,0;285,1;290,0;314,1;321,0;468,-1; 2:5,13,9,Times,0,12,0,0,0;4,13,10,Courier,1,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Problem 9 :[font = text; inactive; preserveAspect; endGroup] Consider the predator-prey model: x'[t] == x[t]( 1 - y[t]) y'[t] == y[t](-2 + x[t]) where x[t] represents the population of the prey and y[t] represents the population of the predator. a) Find all critical points of this system. At each critical point, calculate the corresponding linear system and find the eigenvalues of the coefficient matrix. Then identify the type and stability of the critical point. b) Plot the vector field using PlotVectorField[] with the options ScaleFunction->(Log[50#+1]&) and Frame->True . Choose a representative sample of several initial points {x0,y0} in the first quadrant. Use NDSolve[] to obtain numerical solutions through these points and plot the corresponding trajectories on a single graph. Indicate what happens to the trajectories for increasing t. Explain how the populations vary for initial points close to the critical point inside the first quadrant. Superimpose the graphs of one pair x[t], y[t] as a functions of time t to see how the populations increa/decrease with respect to each other. ;[s] 30:0,0;35,1;94,0;100,1;104,0;147,1;151,0;194,1;195,2;197,0;418,2;420,0;450,1;467,0;484,1;513,0;518,1;529,0;589,1;596,0;624,1;633,0;801,1;802,0;946,1;950,0;952,1;956,0;980,1;981,0;1053,-1; 3:15,13,9,Times,0,12,0,0,0;13,13,10,Courier,1,12,0,0,0;2,13,9,Times,2,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Solution :[font = input; preserveAspect] Clear[x,y] :[font = input; preserveAspect] F[x_,y_] = x( 1 - y); G[x_,y_] = y(-2 + x); :[font = text; inactive; preserveAspect] a) The critical points are: ;[s] 2:0,0;2,1;29,-1; 2:1,13,9,Times,2,12,0,0,0;1,13,9,Times,0,12,0,0,0; :[font = input; preserveAspect; startGroup] cp = Solve[{F[x,y]==0, G[x,y]==0}] :[font = output; output; inactive; preserveAspect; endGroup] {{x -> 0, y -> 0}, {x -> 2, y -> 1}} ;[o] {{x -> 0, y -> 0}, {x -> 2, y -> 1}} :[font = input; preserveAspect; startGroup] A = {{D[F[x,y],x], D[F[x,y],y]}, {D[G[x,y],x], D[G[x,y],y]}}; A//MatrixForm :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{1 - y, -x}, {y, -2 + x}}] ;[o] 1 - y -x y -2 + x :[font = input; preserveAspect; startGroup] A1 = A/.cp[[1]]; A1//MatrixForm :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{1, 0}, {0, -2}}] ;[o] 1 0 0 -2 :[font = text; inactive; preserveAspect] The eigenvalues are 1 and -2, so the critical point {0,0} is an unstable saddle point. ;[s] 7:0,0;20,1;21,0;26,1;28,0;52,1;57,0;87,-1; 2:4,13,9,Times,0,12,0,0,0;3,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] A2 = A/.cp[[2]]; A2//MatrixForm :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{0, -2}, {1, 0}}] ;[o] 0 -2 1 0 :[font = input; preserveAspect; startGroup] Eigenvalues[A2] :[font = output; output; inactive; preserveAspect; endGroup] {-I*2^(1/2), I*2^(1/2)} ;[o] {-I Sqrt[2], I Sqrt[2]} :[font = text; inactive; preserveAspect] The eigenvalues are imaginary, so the critical point {2,1} is a center or a spiral point and has indeterminate stability. ;[s] 3:0,0;53,1;58,0;122,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = text; inactive; preserveAspect] b) ;[s] 1:0,0;3,-1; 1:1,13,9,Times,2,12,0,0,0; :[font = input; preserveAspect; startGroup] v = PlotVectorField[{F[x,y],G[x,y]},{x,0,7},{y,0,5.25}, ScaleFunction->(Log[50#+1]&), Frame->True,AspectRatio->1] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 282] %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.130652 0.0238095 0.17322 [ [(0)] .02381 0 0 2 Msboxa [(1)] .15446 0 0 2 Msboxa [(2)] .28511 0 0 2 Msboxa [(3)] .41577 0 0 2 Msboxa [(4)] .54642 0 0 2 Msboxa [(5)] .67707 0 0 2 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.20098 L .6683 .23238 L .58642 .2986 L .48612 .35866 L .4346 .38223 L .40946 .39184 L .38501 .39989 L .36145 .40631 L .3389 .41111 L .32805 .4129 L .31748 .41429 L .30722 .4153 L .29726 .41593 L .28761 .41618 L Mistroke .27827 .41607 L .26925 .41562 L .26053 .41483 L .25212 .41372 L .24402 .4123 L .22871 .4086 L .21457 .40385 L .20153 .39817 L .17854 .38447 L .15921 .3684 L .14305 .35073 L .11834 .31307 L .109 .29402 L .10124 .27527 L .08949 .23954 L .08153 .20696 L .07864 .19201 L .07633 .17797 L .07455 .16483 L .07321 .15258 L .07269 .14677 L .07227 .14118 L .07194 .13579 L .07181 .13317 L .0717 .13059 L .0716 .12807 L .07153 .12559 L .07148 .12317 L .07145 .12078 L .07143 .11845 L .07143 .11616 L .07146 .11391 L .07149 .11171 L .07155 .10955 L .07162 .10743 L .07171 .10536 L .07182 .10332 L .0724 .09558 L .07279 .09194 L .07323 .08844 L .0743 .08186 L .07559 .07581 L .0771 .07024 L .08077 .0604 L .08292 .05608 L .08529 .0521 L .09068 .0451 L .09696 .03921 L .10417 .03426 L .11236 .0301 L Mistroke .12162 .02661 L .132 .0237 L .14361 .02127 L .15655 .01927 L .17093 .01763 L .18687 .01631 L .19547 .01575 L .20451 .01527 L .224 .01449 L .23448 .0142 L .24549 .01396 L .25703 .01379 L .26301 .01372 L .26605 .0137 L .26913 .01367 L .27225 .01365 L .2754 .01364 L .27859 .01363 L .28182 .01362 L .28508 .01362 L .28838 .01362 L .29172 .01363 L .2951 .01364 L .29852 .01365 L .30197 .01367 L .309 .01372 L .31619 .01378 L .32354 .01387 L .33874 .0141 L .3546 .01442 L .3884 .01535 L .42498 .01676 L .46429 .01881 L .50614 .02176 L .55003 .02596 L .57249 .0287 L .59506 .032 L .63963 .04076 L .66096 .04655 L .68108 .05357 L .69941 .06209 L .70769 .06701 L .71525 .07241 L .72199 .07836 L .72779 .08489 L .73029 .0884 L .73252 .09206 L .73445 .0959 L .73606 .09991 L .73734 .1041 L Mistroke .73828 .10848 L .73885 .11305 L .73903 .11782 L .73882 .12279 L .7382 .12796 L .73715 .13334 L .73565 .13894 L .72493 .16342 L .71658 .17691 L .70615 .19119 L .64382 .25445 L .55396 .32014 L .50313 .34979 L .45141 .37509 L .40099 .39478 L .37683 .40227 L .3536 .40813 L .34238 .41046 L .33143 .41237 L .32078 .41389 L .31042 .41502 L .30036 .41577 L .29061 .41614 L .28118 .41614 L .27205 .4158 L .26324 .41511 L .25473 .4141 L .24654 .41278 L .23864 .41116 L .22374 .40708 L .20998 .402 L .19731 .39602 L .18566 .38927 L .16519 .37389 L .14805 .35667 L .13373 .3383 L .11188 .30026 L .09679 .26297 L .08646 .22822 L .07952 .19682 L Mfstroke P P p p .004 w .10714 .11772 m .10715 .11582 L .10719 .11395 L .10724 .11211 L .10732 .1103 L .10754 .10678 L .10768 .10506 L .10784 .10337 L .10869 .09689 L .10922 .09382 L .10984 .09085 L .11304 .08001 L .11506 .07515 L .11737 .07065 L .12281 .06261 L .12935 .05572 L .13701 .04985 L .14582 .04487 L .15583 .04067 L .16709 .03714 L .17968 .03422 L .19367 .03183 L .20122 .03082 L .20915 .02993 L .21747 .02915 L .22619 .02848 L .23533 .02791 L .2449 .02745 L .24984 .02726 L .2549 .0271 L .26006 .02696 L .26534 .02684 L .26802 .0268 L .27073 .02676 L .27347 .02672 L .27624 .0267 L .27903 .02668 L .28186 .02667 L .28471 .02666 L .28759 .02666 L .2905 .02667 L .29344 .02668 L .29641 .02671 L .29941 .02674 L .3055 .02682 L .3117 .02693 L .31802 .02707 L .32445 .02725 L .33767 .0277 L .35135 .0283 L Mistroke .36547 .02906 L .39495 .03113 L .42587 .03409 L .4578 .0382 L .49001 .04382 L .50591 .04735 L .52145 .05145 L .55055 .06177 L .56356 .06819 L .57513 .07562 L .58486 .08419 L .5889 .08895 L .59233 .09403 L .59509 .09946 L .5962 .10231 L .59712 .10525 L .59785 .10829 L .59837 .11141 L .59868 .11464 L .59877 .11796 L .59865 .12137 L .59829 .12488 L .5977 .12848 L .59687 .13219 L .59446 .13987 L .59103 .14793 L .581 .16506 L .56668 .18337 L .52571 .22194 L .47139 .25936 L .40992 .29045 L .37873 .30217 L .3634 .30691 L .34834 .31086 L .33365 .31401 L .31939 .31635 L .31243 .31722 L .3056 .31789 L .2989 .31837 L .29234 .31866 L .28591 .31876 L .27962 .31867 L .27348 .31841 L .26748 .31796 L .26163 .31735 L .25593 .31657 L .24497 .31454 L .23459 .31192 L .22481 .30874 L .20693 .30096 L Mistroke .1912 .29158 L .17745 .28099 L .15516 .25746 L .14626 .24508 L .13863 .2326 L .12661 .20798 L .11812 .18459 L .11496 .17355 L .11242 .16299 L .11044 .15296 L .10896 .14345 L .1084 .13889 L .10794 .13447 L .10759 .13018 L .10735 .12601 L .10726 .12398 L .1072 .12198 L .10716 .12001 L .10715 .11807 L .10715 .11617 L .10718 .11429 L .10723 .11245 L .10731 .11063 L .1074 .10885 L .10752 .1071 L .10781 .10368 L .10819 .10037 L .10864 .09718 L .10978 .09112 L .11122 .08548 L .11295 .08024 L .11497 .07537 L .11983 .06666 L .12267 .06278 L .12579 .05919 L .13286 .05281 L .14107 .04737 L .15044 .04278 L .16104 .0389 L .17293 .03568 L .18618 .03302 L .20086 .03087 L .20878 .02997 L .21708 .02918 L .22579 .02851 L .23491 .02794 L .24445 .02747 L .25443 .02711 L .25959 .02697 L .26485 .02685 L Mistroke .27024 .02676 L .27297 .02673 L .27573 .0267 L .27852 .02668 L .28134 .02667 L .28419 .02666 L .28706 .02666 L .28997 .02667 L .2929 .02668 L .29587 .0267 L .29886 .02673 L .30494 .02681 L .31113 .02692 L .32386 .02723 L .33706 .02768 L .35071 .02827 L .37934 .02995 L .40955 .03242 L .44102 .03589 L .47319 .04067 L .50519 .04717 L .52075 .05125 L .53574 .05599 L .54992 .06149 L .563 .06788 L .57464 .07526 L .5798 .07937 L .58446 .08378 L .58856 .0885 L .59205 .09355 L .59355 .09621 L .59487 .09895 L .59601 .10178 L .59697 .10471 L .59773 .10773 L .59829 .11084 L .59864 .11404 L .59878 .11734 L .59869 .12074 L .59837 .12423 L .59783 .12782 L .59704 .1315 L .59472 .13915 L .59139 .14717 L .58701 .15555 L .58156 .16425 L .56743 .18251 L .54909 .2016 L .50113 .24025 L .44251 .27529 L Mistroke .41135 .28984 L .38015 .3017 L .36479 .30651 L .34971 .31053 L .33498 .31375 L .32067 .31617 L .3137 .31708 L .30684 .31779 L .30012 .3183 L .29353 .31862 L .28707 .31876 L .28076 .3187 L .27459 .31847 L .26857 .31806 L .26269 .31748 L .25696 .31673 L .24594 .31475 L .23552 .31218 L .22568 .30906 L .2077 .30136 L .19188 .29204 L .17804 .28149 L .16601 .27005 L .14664 .24565 L .1324 .22075 L .12217 .19663 L .11509 .17404 L .11252 .16347 L .11051 .1534 L .10901 .14387 L .10844 .1393 L .10798 .13487 L .10762 .13056 L .10748 .12846 L .10737 .12639 L .10727 .12435 L .10721 .12234 L .10716 .12037 L .10714 .11843 L .10715 .11651 L .10717 .11463 L .10722 .11278 L .10729 .11096 L .10749 .10741 L .10763 .10568 L .10778 .10398 L .10859 .09747 L .10912 .09437 L .10972 .09139 L .11115 .08573 L Mistroke .11287 .08047 L Mfstroke P P p p .004 w .14286 .11772 m .14287 .1162 L .14291 .1147 L .14297 .11321 L .14305 .11175 L .14328 .10888 L .14343 .10748 L .14361 .1061 L .14452 .10075 L .14511 .0982 L .14577 .09572 L .14926 .08654 L .15147 .08237 L .15399 .07847 L .15994 .07143 L .16709 .06534 L .17544 .0601 L .18499 .05565 L .19576 .05191 L .2078 .04882 L .21429 .04751 L .22111 .04635 L .23573 .04445 L .24354 .04371 L .25168 .0431 L .26016 .04264 L .26452 .04246 L .26897 .04231 L .27349 .0422 L .27579 .04216 L .27811 .04212 L .28044 .0421 L .2828 .04208 L .28517 .04207 L .28757 .04208 L .28998 .04209 L .29242 .04211 L .29488 .04214 L .29735 .04218 L .30236 .04229 L .30744 .04244 L .31783 .04286 L .32849 .04344 L .33941 .04421 L .35055 .04518 L .37339 .04776 L .39664 .05135 L .41978 .05615 L .44211 .06242 L .45269 .06621 L Mistroke .46269 .0705 L .47197 .07533 L .48034 .08075 L .48762 .0868 L .49079 .09008 L .49361 .09354 L .49606 .09717 L .49811 .10099 L .49898 .10297 L .49974 .10499 L .50039 .10707 L .50092 .10918 L .50133 .11135 L .50162 .11356 L .50179 .11582 L .50183 .11812 L .50175 .12047 L .50153 .12287 L .50118 .12531 L .50069 .12779 L .49929 .13288 L .49733 .13813 L .4948 .14353 L .49168 .14907 L .47341 .17217 L .46094 .18396 L .44647 .19559 L .41267 .21729 L .39404 .22682 L .37477 .23513 L .35523 .24203 L .33581 .24736 L .32624 .24941 L .31681 .25105 L .31216 .25171 L .30756 .25226 L .30301 .25271 L .29851 .25306 L .29407 .2533 L .28969 .25344 L .28537 .25349 L .28112 .25343 L .27694 .25327 L .27282 .25302 L .26877 .25268 L .26479 .25224 L .25706 .25111 L .24963 .24963 L .2425 .24783 L .23567 .24573 L Mistroke .22293 .24072 L .2114 .23477 L .20102 .22805 L .19175 .22074 L .17626 .20491 L .16437 .18834 L .15555 .1718 L .15214 .16374 L .14934 .15587 L .14709 .14825 L .14535 .1409 L .14467 .13733 L .1441 .13383 L .14364 .13042 L .14329 .12708 L .14315 .12544 L .14304 .12382 L .14296 .12222 L .1429 .12064 L .14287 .11908 L .14286 .11754 L .14287 .11602 L .14291 .11452 L .14297 .11304 L .14306 .11158 L .14317 .11013 L .1433 .10871 L .14405 .10323 L .14455 .1006 L .14514 .09805 L .14657 .09317 L .14832 .08859 L .15039 .08429 L .15277 .08027 L .15545 .07651 L .16172 .06973 L .16919 .06387 L .17785 .05885 L .18772 .05459 L .19883 .05103 L .2112 .04811 L .21786 .04688 L .22486 .04579 L .23218 .04484 L .23983 .04404 L .24781 .04337 L .25613 .04284 L .26042 .04263 L .26479 .04245 L .26924 .0423 L Mistroke .27377 .04219 L .27607 .04215 L .27838 .04212 L .28072 .04209 L .28308 .04208 L .28546 .04207 L .28786 .04208 L .29028 .04209 L .29271 .04211 L .29517 .04214 L .29765 .04219 L .30266 .0423 L .30775 .04245 L .31291 .04264 L .32344 .04315 L .33425 .04383 L .35654 .04578 L .37953 .04861 L .40281 .05249 L .42581 .05765 L .44778 .06437 L .45807 .06842 L .46771 .07299 L .47653 .07812 L .48435 .08387 L .49097 .09029 L .4962 .0974 L .49822 .10123 L .49908 .10321 L .49983 .10524 L .50046 .10732 L .50098 .10944 L .50138 .11161 L .50165 .11383 L .50181 .11609 L .50183 .1184 L .50173 .12076 L .50149 .12316 L .50113 .1256 L .50062 .12809 L .49919 .13319 L .4972 .13845 L .49463 .14386 L .48774 .15506 L .47852 .16665 L .46706 .17842 L .43808 .2016 L .4029 .22248 L .38388 .23139 L .36442 .23898 L Mistroke .3449 .24506 L .33523 .2475 L .32566 .24953 L .31625 .25113 L .3116 .25178 L .30701 .25232 L .30246 .25276 L .29797 .25309 L .29354 .25332 L .28917 .25346 L .28486 .25348 L .28061 .25342 L .27644 .25325 L .27233 .25299 L .26432 .25219 L .26043 .25165 L .2566 .25103 L .24207 .24771 L .23527 .2456 L .22876 .2432 L .20575 .23129 L .19597 .22424 L .18726 .21667 L .17278 .20055 L .16175 .18392 L .15366 .16751 L .15058 .15954 L .14808 .1518 L .1461 .14431 L .14463 .13711 L .14406 .13363 L .14361 .13021 L .14327 .12688 L .14314 .12524 L .14303 .12362 L .14295 .12203 L .14289 .12045 L .14286 .11889 L .14286 .11735 L .14288 .11583 L .14292 .11434 L .14298 .11286 L .14307 .1114 L .14318 .10996 L .14332 .10854 L .14365 .10576 L .14408 .10307 L .14459 .10044 L .14586 .09543 L .14746 .09071 L Mistroke .14938 .08628 L .15161 .08213 L .157 .07462 L .16358 .06809 L .17137 .06246 L Mfstroke P P p p .004 w .17857 .11772 m .17858 .11658 L .17862 .11544 L .17867 .11432 L .17875 .11321 L .17897 .11103 L .17928 .10889 L .17967 .1068 L .18015 .10476 L .18134 .10083 L .18286 .09709 L .18468 .09355 L .1868 .0902 L .18923 .08704 L .19494 .08129 L .20178 .07627 L .20973 .07195 L .21877 .06829 L .22886 .06527 L .23431 .06399 L .24 .06287 L .24595 .0619 L .25215 .06108 L .25859 .0604 L .26526 .05988 L .26868 .05968 L .27216 .05951 L .27569 .05938 L .27748 .05933 L .27928 .05929 L .28109 .05926 L .28291 .05924 L .28475 .05923 L .2866 .05923 L .28845 .05924 L .29033 .05926 L .29221 .05929 L .2941 .05933 L .29792 .05944 L .30178 .0596 L .30962 .06003 L .31758 .06065 L .32565 .06145 L .34196 .06364 L .35826 .0667 L .37417 .07073 L .3892 .07587 L .40279 .08223 L .40884 .08591 L .41427 .08994 L Mistroke .41899 .09434 L .4229 .09911 L .42453 .10164 L .42592 .10426 L .42706 .10697 L .42754 .10836 L .42795 .10977 L .42829 .1112 L .42857 .11266 L .42877 .11414 L .42891 .11563 L .42897 .11715 L .42896 .11869 L .42888 .12025 L .42872 .12183 L .42818 .12504 L .42779 .12668 L .42733 .12832 L .4247 .13506 L .42291 .13851 L .42081 .14199 L .41567 .14902 L .4018 .16306 L .38371 .17633 L .36239 .18796 L .35091 .19289 L .33911 .1971 L .32714 .20054 L .32115 .20195 L .31518 .20315 L .30925 .20413 L .30337 .2049 L .30045 .2052 L .29755 .20545 L .29468 .20564 L .29182 .20578 L .28899 .20587 L .28619 .20591 L .28341 .20589 L .28066 .20582 L .27794 .2057 L .27525 .20553 L .26997 .20504 L .26738 .20472 L .26482 .20436 L .25982 .20349 L .25027 .20122 L .24134 .19831 L .23306 .19483 L .21847 .18647 L Mistroke .20645 .17674 L .19685 .16621 L .18949 .15534 L .18659 .1499 L .18417 .14452 L .18221 .13921 L .1807 .13402 L .1801 .13147 L .1796 .12896 L .1792 .12648 L .1789 .12405 L .17879 .12284 L .1787 .12165 L .17863 .12047 L .17859 .1193 L .17857 .11814 L .17858 .11699 L .1786 .11586 L .17865 .11473 L .17872 .11362 L .17881 .11252 L .17906 .11035 L .17939 .10823 L .17981 .10615 L .1809 .10215 L .1823 .09834 L .18402 .09473 L .18604 .09132 L .19098 .08506 L .19707 .07956 L .20428 .07477 L .21259 .07067 L .22197 .06722 L .23242 .06441 L .24389 .06221 L .25001 .06134 L .25636 .06062 L .26296 .06004 L .26634 .05981 L .26978 .05962 L .27328 .05947 L .27504 .0594 L .27682 .05935 L .27862 .05931 L .28042 .05927 L .28224 .05925 L .28407 .05924 L .28592 .05923 L .28777 .05924 L .28964 .05925 L Mistroke .29152 .05928 L .29341 .05932 L .29531 .05936 L .29914 .05949 L .30302 .05966 L .30693 .05987 L .31485 .06042 L .32288 .06115 L .331 .06208 L .34735 .06454 L .36357 .06791 L .37924 .07229 L .39387 .07782 L .4006 .08105 L .40684 .08461 L .41249 .08852 L .41747 .0928 L .42167 .09744 L .425 .10246 L .42631 .10511 L .42738 .10785 L .42818 .11068 L .42848 .11212 L .42871 .11359 L .42887 .11508 L .42896 .1166 L .42898 .11813 L .42892 .11968 L .42879 .12125 L .42858 .12284 L .4283 .12445 L .42795 .12608 L .427 .12938 L .42574 .13274 L .42228 .13961 L .41757 .14662 L .41162 .15368 L .39627 .16756 L .37699 .18039 L .3662 .18613 L .35487 .19128 L .34316 .19575 L .33124 .19946 L .31926 .20235 L .31329 .20348 L .30737 .2044 L .30444 .20478 L .30151 .2051 L .29861 .20537 L .29572 .20558 L Mistroke .29286 .20574 L .29002 .20585 L .28721 .2059 L .28442 .20591 L .28166 .20585 L .27893 .20575 L .27623 .2056 L .27356 .2054 L .27092 .20515 L .26832 .20484 L .26322 .2041 L .25826 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Mistroke .27096 .06319 L .27346 .06314 L .27598 .0631 L .27853 .06307 L .28111 .06305 L .28372 .06304 L .28635 .06304 L .28901 .06305 L .29169 .06307 L .2944 .0631 L .29714 .06314 L .30269 .06326 L .30836 .06342 L .32 .06388 L .33207 .06454 L .34456 .06541 L .37074 .06788 L .39837 .07151 L .42715 .07662 L .45657 .08365 L .48584 .09322 L .50007 .09922 L .51378 .1062 L .52675 .11429 L .53871 .12369 L .54935 .13455 L .55407 .1406 L .55833 .14708 L .56208 .15403 L .56528 .16147 L .56665 .16537 L .56786 .16941 L .5689 .17358 L .56978 .17788 L .57047 .18232 L .57098 .1869 L .5713 .19161 L .57143 .19647 L .57135 .20147 L .57106 .20661 L .57056 .21189 L .56984 .21731 L .56772 .22857 L .56467 .24037 L .56066 .25269 L .55568 .2655 L .54276 .29236 L .52598 .32046 L .48212 .37733 L .42852 .42889 L Mistroke .40002 .4503 L .37148 .46775 L .35743 .47482 L .34364 .48074 L .33017 .48548 L .31709 .48903 L .31071 .49037 L .30444 .49141 L .29829 .49217 L .29226 .49265 L .28636 .49284 L .28058 .49276 L .27494 .49242 L .26943 .49182 L .26406 .49096 L .25882 .48986 L .24874 .48695 L .23921 .48317 L .23021 .47857 L .21376 .46724 L .19929 .45354 L .18664 .43801 L .17564 .42118 L .15792 .38527 L .14489 .34863 L .13554 .31313 L .12907 .2799 L .12672 .26434 L .12488 .24954 L .12351 .23551 L .12299 .22879 L .12256 .22226 L .12224 .21593 L .12211 .21283 L .122 .20978 L .12192 .20679 L .12186 .20383 L .12182 .20093 L .1218 .19807 L .1218 .19525 L .12183 .19249 L .12187 .18976 L .12193 .18709 L .12212 .18186 L .12224 .17932 L .12238 .17682 L .12313 .16723 L .12361 .16268 L .12416 .15828 L .12546 .14996 L Mistroke .12703 .14222 L Mfstroke P P p p .004 w .15446 .19703 m .15447 .19479 L .15451 .19258 L .15456 .1904 L .15464 .18824 L .15485 .18402 L .15499 .18196 L .15515 .17992 L .15598 .17206 L .15652 .1683 L .15713 .16466 L .16031 .15114 L .16234 .14501 L .16465 .13928 L .17009 .12892 L .17663 .11995 L .18426 .11225 L .19299 .10569 L .20285 .10019 L .21385 .09565 L .21979 .09372 L .22603 .09201 L .2394 .08921 L .24654 .08812 L .25399 .08723 L .26174 .08655 L .26573 .08628 L .2698 .08607 L .27394 .0859 L .27604 .08584 L .27815 .08579 L .28029 .08575 L .28245 .08573 L .28462 .08572 L .28681 .08572 L .28902 .08574 L .29125 .08577 L .29349 .08582 L .29576 .08588 L .30034 .08604 L .30498 .08625 L .31448 .08687 L .32423 .08773 L .33422 .08887 L .34441 .09029 L .3653 .09409 L .38656 .09936 L .40773 .10643 L .42815 .11566 L .43782 .12124 L Mistroke .44697 .12755 L .45545 .13465 L .46311 .14262 L .46977 .15153 L .47267 .15636 L .47525 .16145 L .47749 .1668 L .47937 .17241 L .48016 .17532 L .48085 .1783 L .48145 .18135 L .48193 .18447 L .48231 .18765 L .48258 .19091 L .48273 .19423 L .48277 .19762 L .48269 .20108 L .48249 .2046 L .48217 .20819 L .48172 .21184 L .48045 .21933 L .47865 .22706 L .47634 .23501 L .47348 .24315 L .45677 .27715 L .44537 .29449 L .43214 .31161 L .40123 .34354 L .38419 .35756 L .36656 .3698 L .34869 .37994 L .33093 .38779 L .32217 .39081 L .31355 .39321 L .3093 .39418 L .30509 .395 L .30093 .39566 L .29682 .39617 L .29276 .39653 L .28875 .39674 L .2848 .3968 L .28091 .39672 L .27709 .39649 L .27332 .39612 L .26962 .39561 L .26598 .39497 L .25891 .3933 L .25211 .39112 L .24559 .38847 L .23934 .38539 L Mistroke .22769 .37801 L .21714 .36926 L .20766 .35938 L .19918 .34861 L .18501 .32533 L .17413 .30094 L .16607 .27661 L .16295 .26474 L .16039 .25317 L .15833 .24195 L .15674 .23113 L .15612 .22588 L .15559 .22074 L .15518 .21571 L .15486 .2108 L .15473 .20839 L .15463 .206 L .15455 .20365 L .1545 .20132 L .15447 .19903 L .15446 .19676 L .15448 .19452 L .15451 .19231 L .15457 .19014 L .15465 .18799 L .15475 .18587 L .15487 .18378 L .15555 .1757 L .15601 .17183 L .15655 .16808 L .15786 .16091 L .15946 .15417 L .16135 .14784 L .16353 .14192 L .16598 .13639 L .17172 .12641 L .17854 .11779 L .18646 .1104 L .19549 .10414 L .20565 .0989 L .21696 .0946 L .22306 .09279 L .22946 .09119 L .23615 .08979 L .24315 .08861 L .25045 .08763 L .25806 .08684 L .26198 .08653 L .26597 .08627 L .27004 .08605 L Mistroke .27419 .08589 L .27629 .08583 L .27841 .08578 L .28055 .08575 L .28271 .08573 L .28488 .08572 L .28707 .08572 L .28929 .08574 L .29152 .08578 L .29376 .08582 L .29603 .08588 L .30061 .08605 L .30527 .08627 L .30999 .08655 L .31962 .0873 L .3295 .0883 L .34989 .09117 L .37092 .09533 L .3922 .10105 L .41324 .10864 L .43334 .11853 L .44275 .12449 L .45156 .13121 L .45963 .13876 L .46678 .14722 L .47283 .15666 L .47762 .16713 L .47947 .17276 L .48025 .17568 L .48093 .17867 L .48151 .18172 L .48198 .18485 L .48235 .18804 L .4826 .1913 L .48274 .19464 L .48277 .19803 L .48267 .2015 L .48246 .20503 L .48212 .20863 L .48166 .21229 L .48035 .21979 L .47853 .22754 L .47618 .2355 L .46988 .25198 L .46145 .26902 L .45097 .28635 L .42446 .32045 L .39229 .35118 L .37489 .36429 L .35709 .37545 L Mistroke .33924 .3844 L .3304 .38799 L .32165 .39097 L .31304 .39334 L .30879 .39429 L .30459 .39509 L .30043 .39573 L .29633 .39622 L .29227 .39656 L .28827 .39675 L .28433 .3968 L .28045 .3967 L .27663 .39645 L .27287 .39607 L .26555 .39489 L .26199 .3941 L .25849 .39318 L .2452 .3883 L .23898 .38519 L .23303 .38167 L .21198 .36415 L .20304 .35377 L .19507 .34263 L .18182 .31891 L .17174 .29444 L .16435 .27028 L .16153 .25856 L .15924 .24717 L .15743 .23616 L .15608 .22557 L .15557 .22043 L .15515 .21541 L .15484 .21051 L .15472 .2081 L .15462 .20572 L .15455 .20337 L .1545 .20104 L .15447 .19875 L .15446 .19649 L .15448 .19425 L .15452 .19205 L .15458 .18987 L .15466 .18773 L .15476 .18561 L .15488 .18352 L .15519 .17944 L .15558 .17546 L .15604 .17161 L .15721 .16422 L .15867 .15728 L Mistroke .16043 .15076 L .16247 .14466 L .16739 .13361 L .17342 .124 L .18053 .11572 L Mfstroke P P p p .004 w .18712 .19703 m .18714 .19535 L .18717 .19368 L .18722 .19203 L .18729 .1904 L .18749 .18718 L .18777 .18404 L .18813 .18097 L .18857 .17796 L .18966 .17217 L .19104 .16667 L .19271 .16146 L .19465 .15653 L .19687 .15189 L .20209 .14343 L .20835 .13604 L .21562 .12967 L .22389 .12429 L .23312 .11985 L .2381 .11797 L .24331 .11632 L .24875 .11489 L .25442 .11368 L .26031 .11269 L .26641 .11192 L .26954 .11162 L .27272 .11138 L .27595 .11119 L .27758 .11112 L .27923 .11106 L .28088 .11101 L .28255 .11098 L .28423 .11097 L .28592 .11097 L .28762 .11098 L .28933 .11101 L .29105 .11106 L .29279 .11112 L .29628 .11128 L .29981 .11151 L .30697 .11215 L .31426 .11305 L .32163 .11422 L .33655 .11745 L .35146 .12195 L .36601 .12789 L .37976 .13544 L .39219 .1448 L .39772 .15022 L .40268 .15616 L Mistroke .407 .16263 L .41058 .16965 L .41207 .17336 L .41334 .17722 L .41439 .18121 L .41482 .18325 L .4152 .18533 L .41551 .18744 L .41576 .18958 L .41595 .19175 L .41607 .19396 L .41613 .1962 L .41612 .19846 L .41605 .20076 L .4159 .20308 L .41541 .2078 L .41505 .21021 L .41463 .21263 L .41223 .22255 L .41059 .22762 L .40867 .23274 L .40397 .24309 L .39128 .26374 L .37474 .28328 L .35524 .30038 L .34474 .30763 L .33394 .31383 L .323 .31889 L .31753 .32097 L .31207 .32273 L .30664 .32418 L .30126 .32531 L .29859 .32575 L .29594 .32612 L .29331 .3264 L .2907 .32661 L .28811 .32674 L .28555 .32679 L .28301 .32677 L .28049 .32666 L .278 .32649 L .27554 .32624 L .27071 .32551 L .26835 .32505 L .26601 .32451 L .26143 .32323 L .25269 .31989 L .24453 .31561 L .23696 .31049 L .22362 .29818 L Mistroke .21262 .28387 L .20384 .26838 L .19711 .25239 L .19445 .24439 L .19224 .23646 L .19045 .22866 L .18907 .22101 L .18852 .21726 L .18806 .21356 L .1877 .20992 L .18743 .20634 L .18732 .20457 L .18724 .20282 L .18718 .20108 L .18714 .19935 L .18713 .19765 L .18713 .19596 L .18715 .19429 L .1872 .19263 L .18726 .19099 L .18734 .18937 L .18757 .18618 L .18788 .18306 L .18826 .18001 L .18925 .17412 L .19054 .16852 L .19211 .16321 L .19396 .15818 L .19847 .14898 L .20404 .14087 L .21064 .13383 L .21824 .12779 L .22682 .12273 L .23637 .11859 L .24686 .11535 L .25246 .11407 L .25827 .113 L .2643 .11216 L .2674 .11182 L .27054 .11154 L .27374 .11131 L .27535 .11122 L .27698 .11114 L .27862 .11108 L .28027 .11103 L .28194 .11099 L .28361 .11097 L .2853 .11097 L .287 .11098 L .2887 .111 L Mistroke .29042 .11104 L .29215 .11109 L .29389 .11116 L .2974 .11134 L .30094 .11159 L .30451 .1119 L .31176 .11271 L .31911 .11379 L .32653 .11515 L .34148 .11878 L .35631 .12374 L .37065 .13019 L .38403 .13831 L .39019 .14307 L .39589 .14831 L .40106 .15407 L .40561 .16036 L .40945 .16719 L .4125 .17457 L .4137 .17847 L .41467 .1825 L .41541 .18666 L .41568 .18879 L .41589 .19096 L .41604 .19315 L .41612 .19537 L .41614 .19763 L .41608 .19991 L .41597 .20223 L .41578 .20457 L .41552 .20693 L .41519 .20933 L .41433 .21418 L .41318 .21913 L .41002 .22924 L .4057 .23955 L .40026 .24994 L .38623 .27036 L .36859 .28924 L .35873 .29769 L .34836 .30527 L .33765 .31184 L .32675 .3173 L .31579 .32156 L .31034 .32323 L .30492 .32457 L .30224 .32513 L .29956 .3256 L .29691 .326 L .29427 .32631 L Mistroke .29165 .32655 L .28905 .32671 L .28648 .32679 L .28393 .32679 L .28141 .32671 L .27891 .32656 L .27644 .32634 L .274 .32604 L .27159 .32567 L .26921 .32523 L .26454 .32413 L .26001 .32277 L .25561 .32114 L .24725 .31717 L .23947 .31233 L .2323 .30671 L .21974 .29365 L .20949 .27886 L .2014 .26313 L .19531 .24711 L .19102 .2313 L .18949 .2236 L .18888 .21982 L .18836 .21608 L .18794 .2124 L .1876 .20878 L .18747 .20699 L .18736 .20521 L .18727 .20345 L .1872 .20171 L .18715 .19998 L .18713 .19827 L .18712 .19657 L .18714 .1949 L .18717 .19323 L .18723 .19159 L .18731 .18996 L .18741 .18835 L .18766 .18518 L .18799 .18208 L .18888 .17609 L .19006 .1704 L .19154 .16499 L .19329 .15986 L .19532 .15502 L .20019 .14619 L .2061 .13844 L .21303 .13173 L .22095 .12602 L .22986 .12126 L Mistroke Mfstroke P P p p .004 w .21979 .19703 m .2198 .19591 L .21982 .19479 L .21986 .19368 L .21992 .19258 L .22008 .1904 L .22031 .18826 L .2206 .18616 L .22095 .18409 L .22184 .18007 L .22297 .17621 L .22433 .17252 L .22774 .16567 L .23201 .15955 L .23711 .15418 L .24299 .14957 L .2496 .14573 L .2569 .14267 L .26079 .14143 L .26482 .14039 L .26899 .13955 L .27113 .1392 L .27329 .1389 L .27549 .13866 L .27771 .13846 L .27883 .13838 L .27996 .13832 L .28109 .13827 L .28223 .13823 L .28338 .1382 L .28453 .13819 L .28568 .13819 L .28684 .1382 L .28801 .13823 L .28918 .13827 L .29035 .13832 L .29153 .13839 L .29628 .13879 L .29867 .13907 L .30107 .13941 L .31069 .14134 L .31548 .14266 L .32022 .14422 L .32944 .1481 L .33812 .15302 L .34599 .15903 L .34952 .16245 L .35275 .16615 L .35562 .17012 L .3581 .17437 L Mistroke .36016 .17888 L .36102 .18123 L .36176 .18365 L .36238 .18612 L .36287 .18864 L .36307 .18993 L .36323 .19123 L .36337 .19254 L .36346 .19386 L .36353 .19519 L .36356 .19654 L .36355 .19789 L .36351 .19926 L .36344 .20063 L .36333 .20202 L .36318 .20341 L .363 .20481 L .36253 .20764 L .36192 .21048 L .36027 .21623 L .35804 .222 L .35525 .22773 L .34809 .23888 L .33904 .24916 L .32847 .25809 L .32275 .26191 L .31682 .26522 L .31075 .268 L .3046 .2702 L .30151 .27109 L .29842 .27182 L .29533 .2724 L .2938 .27264 L .29227 .27283 L .29074 .27299 L .28922 .27311 L .2877 .2732 L .2862 .27324 L .2847 .27325 L .28321 .27322 L .28173 .27315 L .28026 .27305 L .27735 .27274 L .27591 .27253 L .27449 .27228 L .26892 .27096 L .26623 .27009 L .2636 .2691 L .25854 .26676 L .25376 .26397 L Mistroke .24929 .26077 L .24128 .25335 L .23456 .24485 L .22915 .23564 L .22502 .22605 L .22341 .2212 L .22211 .21635 L .22111 .21153 L .22071 .20915 L .22039 .20678 L .22014 .20443 L .22003 .20326 L .21995 .2021 L .21989 .20094 L .21984 .1998 L .2198 .19865 L .21979 .19752 L .21979 .19639 L .21981 .19527 L .21984 .19416 L .21989 .19306 L .22004 .19087 L .22014 .18979 L .22025 .18872 L .22087 .18453 L .22127 .18249 L .22173 .18049 L .22417 .17291 L .22574 .16938 L .22753 .16602 L .23176 .15986 L .23681 .15445 L .24265 .1498 L .24923 .14592 L .25278 .14427 L .25649 .14282 L .26036 .14156 L .26438 .14049 L .26853 .13963 L .27282 .13896 L .27501 .13871 L .27723 .1385 L .27947 .13834 L .2806 .13829 L .28174 .13824 L .28288 .13821 L .28403 .13819 L .28518 .13818 L .28634 .13819 L .2875 .13821 L Mistroke .28867 .13825 L .28984 .13829 L .29102 .13835 L .29338 .13852 L .29576 .13873 L .30055 .13933 L .30536 .14016 L .31017 .14121 L .31971 .14404 L .32895 .14786 L .33767 .15272 L .34559 .15867 L .34915 .16206 L .35241 .16573 L .35533 .16968 L .35785 .1739 L .35996 .17838 L .36085 .18072 L .36161 .18312 L .36226 .18557 L .36278 .18809 L .36299 .18937 L .36317 .19066 L .36331 .19196 L .36343 .19328 L .36351 .19461 L .36355 .19595 L .36356 .1973 L .36354 .19866 L .36348 .20003 L .36338 .20141 L .36309 .2042 L .36289 .20561 L .36265 .20702 L .36207 .20986 L .36048 .2156 L .35831 .22137 L .35558 .22711 L .34853 .23829 L .33958 .24863 L .32907 .25765 L .32338 .26152 L .31748 .26489 L .31142 .26773 L .30527 .26999 L .30218 .27091 L .29909 .27168 L .296 .27229 L .29293 .27275 L .2914 .27293 L Mistroke .28988 .27307 L .28836 .27317 L .28685 .27323 L .28535 .27325 L .28385 .27324 L .28237 .27319 L .28089 .2731 L .27943 .27298 L .27798 .27282 L .2751 .27239 L .27228 .27183 L .26952 .27113 L .26416 .26933 L .25907 .26704 L .24976 .26114 L .24556 .25762 L .24168 .25378 L .23489 .24534 L .22941 .23616 L .22715 .2314 L .22521 .22658 L .22357 .22172 L .22224 .21688 L .2212 .21206 L .22045 .20729 L .22018 .20494 L .22008 .20377 L .21998 .2026 L .21991 .20145 L .21985 .20029 L .21981 .19915 L .21979 .19801 L .21979 .19688 L .2198 .19576 L .21982 .19464 L .21987 .19354 L .21993 .19244 L .22 .19134 L .2202 .18919 L .22046 .18707 L .22079 .18498 L .22163 .18093 L .2227 .17703 L .22402 .1733 L .22732 .16638 L .23151 .16017 L .23652 .15472 L .24231 .15003 L .24885 .14611 L .25238 .14444 L Mistroke .25608 .14296 L .26393 .1406 L Mfstroke P P p p .004 w .25245 .19703 m .25246 .19647 L .25247 .19591 L .25249 .19535 L .25253 .19479 L .25262 .19368 L .25276 .19259 L .25293 .19151 L .25314 .19044 L .25367 .18835 L .25434 .18632 L .25515 .18436 L .25718 .18069 L .25972 .17739 L .26274 .17449 L .26619 .17202 L .26806 .17096 L .27001 .17002 L .27416 .16851 L .27634 .16794 L .27745 .1677 L .27858 .1675 L .27972 .16733 L .28086 .16719 L .28144 .16714 L .28202 .16709 L .2826 .16705 L .28319 .16702 L .28377 .16699 L .28436 .16698 L .28495 .16697 L .28554 .16697 L .28613 .16698 L .28673 .167 L .28791 .16706 L .28851 .16711 L .28911 .16716 L .2903 .16729 L .29268 .16766 L .29504 .16816 L .29738 .1688 L .30193 .17049 L .30622 .17273 L .31015 .1755 L .3136 .17878 L .31512 .18059 L .31648 .18252 L .31767 .18456 L .31868 .18669 L .3195 .18892 L Mistroke .32012 .19122 L .32035 .19239 L .32045 .19299 L .32053 .19358 L .3206 .19418 L .32066 .19479 L .3207 .1954 L .32073 .19601 L .32075 .19662 L .32075 .19723 L .32074 .19785 L .32071 .19847 L .32067 .19909 L .32062 .19971 L .32047 .20096 L .32027 .20221 L .32001 .20346 L .31933 .20595 L .31843 .20842 L .31731 .21085 L .31447 .2155 L .31088 .21977 L .30666 .22351 L .30192 .22659 L .2994 .22786 L .29682 .22892 L .29418 .22977 L .29284 .23011 L .2915 .2304 L .29016 .23064 L .28948 .23073 L .28881 .23082 L .28814 .23089 L .28746 .23094 L .28679 .23098 L .28612 .23101 L .28545 .23103 L .28479 .23103 L .28412 .23101 L .28346 .23099 L .28279 .23094 L .28214 .23089 L .28083 .23074 L .27953 .23054 L .27825 .23028 L .27574 .22961 L .27332 .22874 L .27099 .22769 L .26666 .22505 L .26283 .2218 L Mistroke .25955 .21805 L .25688 .21392 L .25482 .20953 L .25403 .20727 L .25341 .20499 L .25294 .2027 L .25276 .20155 L .25262 .20041 L .25257 .19983 L .25253 .19926 L .25249 .19869 L .25247 .19813 L .25246 .19756 L .25245 .197 L .25246 .19643 L .25247 .19587 L .2525 .19531 L .25253 .19476 L .25263 .19365 L .25276 .19256 L .25294 .19148 L .25339 .18935 L .25399 .18729 L .25474 .1853 L .25562 .18338 L .25779 .1798 L .26046 .1766 L .26359 .17381 L .26714 .17146 L .27105 .16958 L .27313 .16883 L .27528 .1682 L .27749 .1677 L .27861 .1675 L .27975 .16733 L .2809 .16719 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.23103 L .28475 .23103 L Mistroke .28408 .23101 L .28342 .23098 L .28275 .23094 L .2821 .23089 L .28079 .23073 L .27949 .23053 L .27695 .22996 L .27448 .22919 L .27211 .22822 L .26983 .22707 L .26562 .22427 L .26193 .22088 L .25881 .21702 L .25629 .21281 L .25526 .21062 L .2544 .20837 L .25369 .2061 L .25314 .20381 L .25293 .20266 L .25275 .20152 L .25262 .20037 L .25257 .1998 L .25252 .19923 L .25249 .19866 L .25247 .19809 L .25245 .19753 L .25245 .19696 L .25246 .1964 L .25247 .19584 L .2525 .19528 L .25253 .19472 L .25263 .19362 L .25277 .19253 L .25294 .19144 L .25315 .19038 L .25368 .18828 L .25436 .18626 L .25518 .1843 L .25613 .18243 L .25843 .17894 L .26122 .17584 L .26446 .17316 L .26811 .17093 L .27007 .16999 L .27211 .16918 L .27423 .16849 L .27641 .16792 L .27752 .16769 L .27865 .16749 L .28093 .16718 L Mistroke Mfstroke P P p p .004 w .28511 .19703 m .28511 .19703 L .28511 .19703 L .28511 .19703 L .28511 .19703 L .28511 .19703 L .28511 .19703 L .28511 .19703 L .28511 .19703 L .28511 .19703 L .28511 .19703 L .28511 .19703 L .28511 .19703 L .28511 .19703 L .28511 .19703 L .28511 .19703 L .28511 .19703 L .28511 .19703 L .28511 .19703 L .28511 .19703 L .28511 .19703 L .28511 .19703 L .28511 .19703 L .28511 .19703 L .28511 .19703 L s P P P P 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = text; inactive; preserveAspect] The trajectories near the critical point {2,1} appear to be closed curves a=which implies that the populations of the predator and prey are periodic functions. Let's plot one pair x[t], y[t] close to the critical point to see how the populations increa/decrease with respect to each other. ;[s] 7:0,0;41,1;46,0;180,1;184,0;186,1;190,0;290,-1; 2:4,13,9,Times,0,12,0,0,0;3,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] Plot[Evaluate[solns[[5]]], {t,0,10}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.0952381 0.0147151 0.167558 [ [(2)] .21429 .01472 0 2 Msboxa [(4)] .40476 .01472 0 2 Msboxa [(6)] .59524 .01472 0 2 Msboxa [(8)] .78571 .01472 0 2 Msboxa [(10)] .97619 .01472 0 2 Msboxa [(0.5)] .01131 .09849 1 0 Msboxa [(1)] .01131 .18227 1 0 Msboxa [(1.5)] .01131 .26605 1 0 Msboxa [(2)] .01131 .34983 1 0 Msboxa [(2.5)] .01131 .43361 1 0 Msboxa [(3)] .01131 .51739 1 0 Msboxa [(3.5)] 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endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- ^*)