(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. 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.81002 .13061 L .81126 .12426 L .8125 .11901 L .81374 .1149 L .81498 .11194 L .81622 .11016 L .81746 .10957 L .8187 .11016 L .81994 .11194 L .82118 .1149 L .82242 .11901 L .82366 .12426 L .8249 .13061 L .82738 .14647 L .83234 .18942 L .8373 .24389 L .84722 .36651 L .85218 .42314 L .85466 .44784 L Mistroke .85714 .46929 L .85962 .48697 L .86086 .49424 L .8621 .50041 L .86334 .50542 L .86458 .50925 L .86582 .51186 L .86706 .51324 L .8683 .51335 L .86954 .51221 L .87078 .5098 L .87202 .50612 L .87326 .5012 L .8745 .49505 L .87698 .47917 L .88194 .43424 L .8869 .37486 L .89683 .233 L .90179 .16253 L .90675 .10054 L .90923 .0745 L .91171 .05257 L .91419 .03526 L .91543 .02848 L .91667 .023 L .91791 .01886 L .91915 .01609 L .92039 .01472 L .92163 .01475 L .92287 .01619 L .92411 .01905 L .92535 .02331 L .92659 .02897 L .92907 .04436 L .93155 .06496 L .93651 .12016 L .97619 .60332 L Mfstroke P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect; startGroup] DSolve[{y''[t] + 9y[t] == Cos[2t], y[0] == 1, y'[0] == 0}, y[t],t]//Simplify :[font = output; output; inactive; preserveAspect; endGroup; endGroup] {{y[t] -> (Cos[2*t] + 4*Cos[3*t])/5}} ;[o] Cos[2 t] + 4 Cos[3 t] {{y[t] -> ---------------------}} 5 :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Problem 2 :[font = text; inactive; preserveAspect; endGroup] a) Find the Laplace transform of the function defined to be f[t] = 0 when t < 1 and f[t] = t^2 - 2t + 1 when t >=1. (Hint: Use the function UnitStep[], which is part of the LaplaceTransform package, to define f[t].) b) Use Theorem 6.3.1 (p.294) to find the inverse Laplace transform of F[s_] = E^(-2s)/(s^2 + s - 2) Compare the result to the answer given by InverseLaplaceTransform[]. ;[s] 22:0,1;3,0;60,2;68,0;74,2;79,0;84,2;103,0;109,2;114,0;140,2;150,0;173,2;189,0;209,2;213,0;216,1;218,0;286,2;319,0;362,2;387,0;389,-1; 3:11,13,9,Times,0,12,0,0,0;2,13,9,Times,2,12,0,0,0;9,13,10,Courier,1,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Solution :[font = input; preserveAspect; startGroup] ?UnitStep :[font = print; inactive; preserveAspect; endGroup] UnitStep[x] is a function that is 1 for x > 0 and 0 for x < 0. UnitStep[x1, x2, ...] is 1 for (x1 > 0) && (x2 > 0) && ... and 0 for (x1 < 0) || (x2 < 0) || ... . :[font = input; preserveAspect] f[t_] = UnitStep[t-1]*(t^2 - 2t + 1); :[font = input; preserveAspect; startGroup] Plot[f[t], {t,0,4}] :[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 0.238095 0.0147151 0.0654004 [ [(1)] .2619 .01472 0 2 Msboxa [(2)] .5 .01472 0 2 Msboxa [(3)] .7381 .01472 0 2 Msboxa [(4)] .97619 .01472 0 2 Msboxa [(2)] .01131 .14552 1 0 Msboxa [(4)] .01131 .27632 1 0 Msboxa [(6)] .01131 .40712 1 0 Msboxa [(8)] .01131 .53792 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 .2619 .01472 m .2619 .02097 L s P [(1)] .2619 .01472 0 2 Mshowa p .002 w .5 .01472 m .5 .02097 L s P [(2)] .5 .01472 0 2 Mshowa p .002 w .7381 .01472 m .7381 .02097 L s P [(3)] .7381 .01472 0 2 Mshowa p .002 w .97619 .01472 m .97619 .02097 L s P [(4)] .97619 .01472 0 2 Mshowa p .001 w .07143 .01472 m .07143 .01847 L s P p .001 w .11905 .01472 m .11905 .01847 L s P p .001 w .16667 .01472 m .16667 .01847 L s P p .001 w .21429 .01472 m .21429 .01847 L s P p .001 w .30952 .01472 m .30952 .01847 L s P p .001 w .35714 .01472 m .35714 .01847 L s P p .001 w .40476 .01472 m .40476 .01847 L s P p .001 w .45238 .01472 m .45238 .01847 L s P p .001 w .54762 .01472 m .54762 .01847 L s P p .001 w .59524 .01472 m .59524 .01847 L s P p .001 w .64286 .01472 m .64286 .01847 L s P p .001 w .69048 .01472 m .69048 .01847 L s P p .001 w .78571 .01472 m .78571 .01847 L s P p .001 w .83333 .01472 m .83333 .01847 L s P p .001 w .88095 .01472 m .88095 .01847 L s P p .001 w .92857 .01472 m .92857 .01847 L s P p .002 w 0 .01472 m 1 .01472 L s P p .002 w .02381 .14552 m .03006 .14552 L s P [(2)] .01131 .14552 1 0 Mshowa p .002 w .02381 .27632 m .03006 .27632 L s P [(4)] .01131 .27632 1 0 Mshowa p .002 w .02381 .40712 m .03006 .40712 L s P [(6)] .01131 .40712 1 0 Mshowa p .002 w .02381 .53792 m .03006 .53792 L s P [(8)] .01131 .53792 1 0 Mshowa p .001 w .02381 .04088 m .02756 .04088 L s P p .001 w .02381 .06704 m .02756 .06704 L s P p .001 w .02381 .0932 m .02756 .0932 L s P p .001 w .02381 .11936 m .02756 .11936 L s P p .001 w .02381 .17168 m .02756 .17168 L s P p .001 w .02381 .19784 m .02756 .19784 L s P p .001 w .02381 .224 m .02756 .224 L s P p .001 w .02381 .25016 m .02756 .25016 L s P p .001 w .02381 .30248 m .02756 .30248 L s P p .001 w .02381 .32864 m .02756 .32864 L s P p .001 w .02381 .3548 m .02756 .3548 L s P p .001 w .02381 .38096 m .02756 .38096 L s P p .001 w .02381 .43328 m .02756 .43328 L s P p .001 w .02381 .45944 m .02756 .45944 L s P p .001 w .02381 .4856 m .02756 .4856 L s P p .001 w .02381 .51176 m .02756 .51176 L s P p .001 w .02381 .56408 m .02756 .56408 L s P p .001 w .02381 .59024 m .02756 .59024 L s P p .001 w .02381 .6164 m .02756 .6164 L s P p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .004 w .02381 .01472 m .06349 .01472 L .10317 .01472 L .14286 .01472 L .18254 .01472 L .22222 .01472 L .24206 .01472 L .25198 .01472 L .25694 .01472 L .25942 .01472 L .26066 .01472 L .2619 .01472 L .26314 .01472 L .26438 .01472 L .26563 .01473 L .26687 .01474 L .26811 .01476 L .26935 .01478 L .27183 .01483 L .27431 .01489 L .27679 .01497 L .28175 .01517 L .28671 .01542 L .29167 .01574 L .30159 .01653 L .31151 .01755 L .32143 .0188 L .34127 .02198 L .36111 .02607 L .38095 .03107 L .42063 .04378 L .46032 .06013 L .5 .08012 L .53968 .10373 L .57937 .13098 L .61905 .16187 L .65873 .19638 L .69841 .23453 L .7381 .27632 L .77778 .32173 L .81746 .37078 L .85714 .42347 L .89683 .47978 L .93651 .53974 L .97619 .60332 L s P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect; startGroup] LaplaceTransform[f[t],t,s] //Simplify :[font = output; output; inactive; preserveAspect; endGroup] 2/(E^s*s^3) ;[o] 2 ----- s 3 E s :[font = text; inactive; preserveAspect] Notice that f[t_] = UnitStep[t-1]*(t-1)^2. By Theorem 6.3.1, LaplaceTransform[f[t],t,s] == E^(-s)LaplaceTransform[t^2] == E^(-s) 2/s^3 ;[s] 4:0,0;12,1;41,0;62,1;163,-1; 2:2,13,9,Times,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0; :[font = text; inactive; preserveAspect] b) ;[s] 2:0,0;2,1;4,-1; 2:1,13,9,Times,2,12,0,0,0;1,13,9,Times,0,12,0,0,0; :[font = input; preserveAspect; startGroup] Apart[1/(s^2 + s - 2)] :[font = output; output; inactive; preserveAspect; endGroup] 1/(3*(-1 + s)) - 1/(3*(2 + s)) ;[o] 1 1 ---------- - --------- 3 (-1 + s) 3 (2 + s) :[font = text; inactive; preserveAspect] By Table 6.2.1, the inverse Laplace transform of these terms is :[font = input; preserveAspect] f[t_] = (1/3)E^t - (1/3)E^(-2t); :[font = text; inactive; preserveAspect] Therefore, by Theorem 6.3.1, the inverse Laplace transform of E^(-2s)/(s^2 + s - 2) is ;[s] 3:0,0;62,1;83,0;88,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] g[t_] = UnitStep[t-2]*f[t-2] :[font = output; output; inactive; preserveAspect; endGroup] (-1/(3*E^(2*(-2 + t))) + E^(-2 + t)/3)*UnitStep[-2 + t] ;[o] -2 + t -1 E (------------- + -------) UnitStep[-2 + t] 2 (-2 + t) 3 3 E :[font = input; preserveAspect; startGroup] Plot[g[t],{t,0,4}] :[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 0.238095 0.0147151 0.23957 [ [(1)] .2619 .01472 0 2 Msboxa [(2)] .5 .01472 0 2 Msboxa [(3)] .7381 .01472 0 2 Msboxa [(4)] .97619 .01472 0 2 Msboxa [(0.5)] .01131 .1345 1 0 Msboxa [(1)] .01131 .25429 1 0 Msboxa [(1.5)] .01131 .37407 1 0 Msboxa [(2)] .01131 .49386 1 0 Msboxa [(2.5)] .01131 .61364 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 .2619 .01472 m .2619 .02097 L s P [(1)] .2619 .01472 0 2 Mshowa p .002 w .5 .01472 m .5 .02097 L s P [(2)] .5 .01472 0 2 Mshowa p .002 w .7381 .01472 m .7381 .02097 L s P [(3)] .7381 .01472 0 2 Mshowa p .002 w .97619 .01472 m .97619 .02097 L s P [(4)] .97619 .01472 0 2 Mshowa p .001 w .07143 .01472 m .07143 .01847 L s P p .001 w .11905 .01472 m .11905 .01847 L s P p .001 w .16667 .01472 m .16667 .01847 L s P p .001 w .21429 .01472 m .21429 .01847 L s P p .001 w .30952 .01472 m .30952 .01847 L s P p .001 w .35714 .01472 m .35714 .01847 L s P p .001 w .40476 .01472 m .40476 .01847 L s P p .001 w .45238 .01472 m .45238 .01847 L s P p .001 w .54762 .01472 m .54762 .01847 L s P p .001 w .59524 .01472 m .59524 .01847 L s P p .001 w .64286 .01472 m .64286 .01847 L s P p .001 w .69048 .01472 m .69048 .01847 L s P p .001 w .78571 .01472 m .78571 .01847 L s P p .001 w .83333 .01472 m .83333 .01847 L s P p .001 w .88095 .01472 m .88095 .01847 L s P p .001 w .92857 .01472 m .92857 .01847 L s P p .002 w 0 .01472 m 1 .01472 L s P p .002 w .02381 .1345 m .03006 .1345 L s P [(0.5)] .01131 .1345 1 0 Mshowa p .002 w .02381 .25429 m .03006 .25429 L s P [(1)] .01131 .25429 1 0 Mshowa p .002 w .02381 .37407 m .03006 .37407 L s P [(1.5)] .01131 .37407 1 0 Mshowa p .002 w .02381 .49386 m .03006 .49386 L s P [(2)] .01131 .49386 1 0 Mshowa p .002 w .02381 .61364 m .03006 .61364 L s P [(2.5)] .01131 .61364 1 0 Mshowa p .001 w .02381 .03867 m .02756 .03867 L s P p .001 w .02381 .06263 m .02756 .06263 L s P p .001 w .02381 .08659 m .02756 .08659 L s P p .001 w .02381 .11054 m .02756 .11054 L s P p .001 w .02381 .15846 m .02756 .15846 L s P p .001 w .02381 .18241 m .02756 .18241 L s P p .001 w .02381 .20637 m .02756 .20637 L s P p .001 w .02381 .23033 m .02756 .23033 L s P p .001 w .02381 .27824 m .02756 .27824 L s P p .001 w .02381 .3022 m .02756 .3022 L s P p .001 w .02381 .32616 m .02756 .32616 L s P p .001 w .02381 .35011 m .02756 .35011 L s P p .001 w .02381 .39803 m .02756 .39803 L s P p .001 w .02381 .42198 m .02756 .42198 L s P p .001 w .02381 .44594 m .02756 .44594 L s P p .001 w .02381 .4699 m .02756 .4699 L s P p .001 w .02381 .51781 m .02756 .51781 L s P p .001 w .02381 .54177 m .02756 .54177 L s P p .001 w .02381 .56573 m .02756 .56573 L s P p .001 w .02381 .58968 m .02756 .58968 L s P p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .004 w .02381 .01472 m .06349 .01472 L .10317 .01472 L .14286 .01472 L .18254 .01472 L .22222 .01472 L .2619 .01472 L .30159 .01472 L .34127 .01472 L .38095 .01472 L .42063 .01472 L .46032 .01472 L .48016 .01472 L .49008 .01472 L .49504 .01472 L .49752 .01472 L .49876 .01472 L .5 .01472 L .50124 .01596 L .50248 .0172 L .50496 .01966 L .50992 .0245 L .51984 .03391 L .53968 .05183 L .57937 .08516 L .61905 .117 L .65873 .1492 L .69841 .18338 L .7381 .22098 L .77778 .26341 L .81746 .31212 L .85714 .36863 L .89683 .43467 L .93651 .51215 L .97619 .60332 L s P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect; startGroup] InverseLaplaceTransform[E^(-2s)/(s^2 + s - 2),s,t] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] (-E^(4 - 2*t)/3 + E^(-2 + t)/3)*UnitStep[-2 + t] ;[o] 4 - 2 t -2 + t -E E (--------- + -------) UnitStep[-2 + t] 3 3 :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Problem 3 :[font = text; inactive; preserveAspect; endGroup] Solve the initial value problems using the Laplace transform and plot the solutions: a) y''[t] + 4y[t] == Sin[t] + UnitStep[t-Pi]Sin[t-Pi], y[0] == 0, y'[0] == 0 b) y''''[t] - y[t] == UnitStep[t-1] - UnitStep[t-2], y[0] == 0, y'[0] == 0, y''[0] == 0, y'''[0] == 0 ;[s] 5:0,0;85,2;87,1;164,2;166,1;268,-1; 3:1,13,9,Times,0,12,0,0,0;2,13,10,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] F[s_] := LaplaceTransform[y[t],t,s] :[font = text; inactive; preserveAspect] a) ;[s] 1:0,0;3,-1; 1:1,13,9,Times,2,12,0,0,0; :[font = input; preserveAspect; startGroup] eqn = LaplaceTransform[ y''[t]+4y[t]==Sin[t]+UnitStep[t-Pi]Sin[t-Pi],t,s] :[font = output; output; inactive; preserveAspect; endGroup] 4*LaplaceTransform[y[t], t, s] + s^2*LaplaceTransform[y[t], t, s] - s*y[0] - Derivative[1][y][0] == (1 + s^2)^(-1) - LaplaceTransform[Sin[t]*UnitStep[-Pi + t], t, s] ;[o] 4 LaplaceTransform[y[t], t, s] + 2 s LaplaceTransform[y[t], t, s] - s y[0] - y'[0] == 1 ------ - LaplaceTransform[Sin[t] UnitStep[-Pi + t], t, s] 2 1 + s :[font = input; preserveAspect; startGroup] tran = Solve[eqn/.{y[0]->0, y'[0]->0},Y[s]] :[font = output; output; inactive; preserveAspect; endGroup] {{LaplaceTransform[y[t], t, s] -> -((-1 + LaplaceTransform[Sin[t]*UnitStep[-Pi + t], t, s] + s^2*LaplaceTransform[Sin[t]*UnitStep[-Pi + t], t, s])/ (4 + 5*s^2 + s^4))}} ;[o] {{LaplaceTransform[y[t], t, s] -> -((-1 + LaplaceTransform[Sin[t] UnitStep[-Pi + t], t, s] + 2 s LaplaceTransform[Sin[t] UnitStep[-Pi + t], t, s]) \ 2 4 / (4 + 5 s + s ))}} :[font = input; preserveAspect; startGroup] soln = InverseLaplaceTransform[Y[s]/.First[tran],s,t] :[font = output; output; inactive; preserveAspect; endGroup] Sin[t]/3 - Sin[2*t]/6 - (4*Cos[t/2]^3*Sin[t/2]* UnitStep[-Pi + t])/3 ;[o] t 3 t 4 Cos[-] Sin[-] UnitStep[-Pi + t] Sin[t] Sin[2 t] 2 2 ------ - -------- - ---------------------------------- 3 6 3 :[font = input; preserveAspect; startGroup] Plot[soln,{t,0,4Pi}] :[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 0.0757881 0.270737 0.768065 [ 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.90179 .50727 L .90427 .51317 L .90675 .51803 L .90923 .52184 L .91047 .52334 L .91171 .52457 L .91295 .52553 L .91419 .52621 L .91543 .52662 L .91667 .52676 L .91791 .52662 L .91915 .52621 L .92039 .52553 L .92163 .52457 L .92411 .52184 L .92659 .51803 L .92907 .51317 L .93155 .50727 L .93651 .49246 L .94643 .45177 L Mistroke .95635 .39875 L .97619 .27074 L Mfstroke P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[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] eqn = LaplaceTransform[ y''''[t]-y[t]==UnitStep[t-1]-UnitStep[t-2],t,s] :[font = output; output; inactive; preserveAspect; endGroup] -LaplaceTransform[y[t], t, s] + s^4*LaplaceTransform[y[t], t, s] - s^3*y[0] - s^2*Derivative[1][y][0] - s*Derivative[2][y][0] - Derivative[3][y][0] == -(1/(E^(2*s)*s)) + 1/(E^s*s) ;[o] -LaplaceTransform[y[t], t, s] + 4 3 2 s LaplaceTransform[y[t], t, s] - s y[0] - s y'[0] - (3) 1 1 s y''[0] - y [0] == -(------) + ---- 2 s s E s E s :[font = input; preserveAspect; startGroup] tran = Solve[eqn/.{y[0]->0, y'[0]->0, y''[0]->0, y'''[0]->0},Y[s]] :[font = output; output; inactive; preserveAspect; endGroup] {{LaplaceTransform[y[t], t, s] -> -((1 - E^s)/(E^(2*s)*s*(-1 + s^4)))}} ;[o] s 1 - E {{LaplaceTransform[y[t], t, s] -> -(----------------)}} 2 s 4 E s (-1 + s ) :[font = input; preserveAspect; startGroup] soln = InverseLaplaceTransform[Y[s]/.First[tran],s,t] :[font = output; output; inactive; preserveAspect; endGroup] (1 - E^(2 - t)/4 - E^(-2 + t)/4 - Cos[2 - t]/2)* UnitStep[-2 + t] + (-1 + E^(1 - t)/4 + E^(-1 + t)/4 + Cos[1 - t]/2)*UnitStep[-1 + t] ;[o] 2 - t -2 + t E E Cos[2 - t] (1 - ------ - ------- - ----------) UnitStep[-2 + t] + 4 4 2 1 - t -1 + t E E Cos[1 - t] (-1 + ------ + ------- + ----------) UnitStep[-1 + t] 4 4 2 :[font = input; preserveAspect; startGroup] Plot[soln,{t,0,4},PlotRange->{0,4}] 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endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Problem 4 :[font = text; inactive; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450; endGroup] Solve the initial value problems using the Laplace transform and plot the solutions: a) y''[t]+4y[t] == DiracDelta[t-Pi]-DiracDelta[t-2Pi], y[0] == 0, y'[0] == 0 b) y''''[t] - y[t] == DiracDelta[t-1], y[0] == 0, y'[0] == 0, y''[0] == 0, y'''[0] == 0 ;[s] 5:0,0;85,1;87,2;164,1;166,2;254,-1; 3:1,13,9,Times,0,12,0,0,0;2,13,9,Times,2,12,0,0,0;2,13,10,Courier,1,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Solution :[font = input; preserveAspect] Clear[t,y] :[font = input; preserveAspect] F[s_] := LaplaceTransform[y[t],t,s] :[font = text; inactive; preserveAspect] a) ;[s] 1:0,0;3,-1; 1:1,13,9,Times,2,12,0,0,0; :[font = input; preserveAspect; startGroup] eqn = LaplaceTransform[ y''[t]+4y[t]==DiracDelta[t-Pi]-DiracDelta[t-2Pi],t,s] :[font = output; output; inactive; preserveAspect; endGroup] 4*LaplaceTransform[y[t], t, s] + s^2*LaplaceTransform[y[t], t, s] - s*y[0] - Derivative[1][y][0] == -E^(-2*Pi*s) + E^(-(Pi*s)) ;[o] 4 LaplaceTransform[y[t], t, s] + 2 s LaplaceTransform[y[t], t, s] - s y[0] - y'[0] == -2 Pi s -(Pi s) -E + E :[font = input; preserveAspect; startGroup] tran = Solve[eqn/.{y[0]->0, y'[0]->0},Y[s]] :[font = output; output; inactive; preserveAspect; endGroup] {{LaplaceTransform[y[t], t, s] -> -((1 - E^(Pi*s))/(E^(2*Pi*s)*(4 + s^2)))}} ;[o] Pi s 1 - E {{LaplaceTransform[y[t], t, s] -> -(----------------)}} 2 Pi s 2 E (4 + s ) :[font = input; preserveAspect; startGroup] soln = InverseLaplaceTransform[Y[s]/.First[tran],s,t] :[font = output; output; inactive; preserveAspect; endGroup] -(Sin[2*(-2*Pi + t)]*UnitStep[-2*Pi + t])/2 + (Sin[2*(-Pi + t)]*UnitStep[-Pi + t])/2 ;[o] -(Sin[2 (-2 Pi + t)] UnitStep[-2 Pi + t]) ----------------------------------------- + 2 Sin[2 (-Pi + t)] UnitStep[-Pi + t] ---------------------------------- 2 :[font = input; preserveAspect; startGroup] Plot[soln,{t,0,3Pi}] :[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 0.101051 0.309017 0.588604 [ [(2)] .22591 .30902 0 2 Msboxa [(4)] .42801 .30902 0 2 Msboxa [(6)] .63011 .30902 0 2 Msboxa [(8)] .83222 .30902 0 2 Msboxa [(-0.4)] .01131 .07358 1 0 Msboxa [(-0.2)] .01131 .1913 1 0 Msboxa [(0.2)] .01131 .42674 1 0 Msboxa [(0.4)] .01131 .54446 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 .22591 .30902 m .22591 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Mfstroke P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[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] eqn = LaplaceTransform[ y''''[t] - y[t] == DiracDelta[t-1],t,s] :[font = output; output; inactive; preserveAspect; endGroup] -LaplaceTransform[y[t], t, s] + s^4*LaplaceTransform[y[t], t, s] - s^3*y[0] - s^2*Derivative[1][y][0] - s*Derivative[2][y][0] - Derivative[3][y][0] == E^(-s) ;[o] -LaplaceTransform[y[t], t, s] + 4 3 2 s LaplaceTransform[y[t], t, s] - s y[0] - s y'[0] - (3) -s s y''[0] - y [0] == E :[font = input; preserveAspect; startGroup] tran = Solve[eqn/.{y[0]->0, y'[0]->0, y''[0] -> 0, y'''[0] -> 0},Y[s]] :[font = output; output; inactive; preserveAspect; endGroup] {{LaplaceTransform[y[t], t, s] -> 1/(E^s*(-1 + s^4))}} ;[o] 1 {{LaplaceTransform[y[t], t, s] -> ------------}} s 4 E (-1 + s ) :[font = input; preserveAspect; startGroup] soln = InverseLaplaceTransform[Y[s]/.First[tran],s,t] :[font = output; output; inactive; preserveAspect; endGroup] (-E^(1 - t)/4 + E^(-1 + t)/4 + Sin[1 - t]/2)*UnitStep[-1 + t] ;[o] 1 - t -1 + t -E E Sin[1 - t] (------- + ------- + ----------) UnitStep[-1 + t] 4 4 2 :[font = input; preserveAspect; startGroup] Plot[soln,{t,0,3},PlotRange->{0,2}] :[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 0.31746 0 0.309017 [ [(0)] .02381 0 0 2 Msboxa [(0.5)] .18254 0 0 2 Msboxa [(1)] .34127 0 0 2 Msboxa [(1.5)] .5 0 0 2 Msboxa [(2)] .65873 0 0 2 Msboxa [(2.5)] .81746 0 0 2 Msboxa [(3)] .97619 0 0 2 Msboxa [(0.25)] .01131 .07725 1 0 Msboxa [(0.5)] .01131 .15451 1 0 Msboxa [(0.75)] .01131 .23176 1 0 Msboxa [(1)] .01131 .30902 1 0 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= 18; rightWrapOffset = 450; endGroup] a) Use the Convolution Theorem 6.6.1 (p.308) to express the inverse Laplace transform of the function H[s_] := 1/((s+1)^2(s^2+4)) in terms of an integral. Evaluate this integral with Integrate[] and compare the final result with InverseLaplaceTransform[H[s],s,t]. b) Express the solution of the initial value problem y''[t] + 4y'[t] + 4y[t] == g[t], y[0]==2, y'[0]==-3 in terms of a convolution integral. ;[s] 13:0,1;3,0;102,2;129,0;183,2;194,0;229,2;262,0;263,2;264,1;266,0;317,2;373,0;409,-1; 3:6,13,9,Times,0,12,0,0,0;2,13,9,Times,2,12,0,0,0;5,13,10,Courier,1,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Solution :[font = text; inactive; preserveAspect] a) H[s] is a product of F[s_] = 1/(s+1)^2 and G[s_] = 1/(s^2+4). Let f[t] and g[t] be the inverse Laplace transforms of F[s] and G[s], respectively. ;[s] 16:0,0;2,1;4,2;8,1;25,2;42,1;47,2;64,1;70,2;74,1;79,2;83,1;121,2;125,1;130,2;134,1;150,-1; 3:1,13,9,Times,2,12,0,0,0;8,13,9,Times,0,12,0,0,0;7,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] f[t_] = InverseLaplaceTransform[1/(s+1)^2,s,t] :[font = output; output; inactive; preserveAspect; endGroup] t/E^t ;[o] t -- t E :[font = input; preserveAspect; startGroup] g[t_] = InverseLaplaceTransform[1/(s^2+4),s,t] :[font = output; output; inactive; preserveAspect; endGroup] Sin[2*t]/2 ;[o] Sin[2 t] -------- 2 :[font = text; inactive; preserveAspect] By the Convolution Theorem, the inverse Laplace transform h[t] of H[s] is the convolution of f[t] and g[t]: ;[s] 9:0,0;59,1;63,0;67,1;71,0;94,1;98,0;103,1;107,0;109,-1; 2:5,13,9,Times,0,12,0,0,0;4,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] Integrate[f[t-u]g[u],{u,0,t}] :[font = output; output; inactive; preserveAspect; endGroup] -(-4 - 10*t)/(50*E^t) + (-4*Cos[2*t] - 3*Sin[2*t])/50 ;[o] -(-4 - 10 t) -4 Cos[2 t] - 3 Sin[2 t] ------------ + ------------------------ t 50 50 E :[font = input; preserveAspect; startGroup] InverseLaplaceTransform[1/((s+1)^2(s^2+4)),s,t] :[font = output; output; inactive; preserveAspect; endGroup] 2/(25*E^t) + t/(5*E^t) + (-2*Cos[2*t] - (3*Sin[2*t])/2)/25 ;[o] 3 Sin[2 t] -2 Cos[2 t] - ---------- 2 t 2 ----- + ---- + ------------------------ t t 25 25 E 5 E :[font = text; inactive; preserveAspect] b) ;[s] 2:0,2;2,1;3,-1; 3:0,13,9,Times,1,12,0,0,0;1,13,9,Times,3,12,0,0,0;1,13,9,Times,2,12,0,0,0; :[font = input; preserveAspect] Clear[t,y,g] :[font = input; preserveAspect] Y[s_] := LaplaceTransform[y[t],t,s]; G[s_] := LaplaceTransform[g[t],t,s] :[font = input; preserveAspect; startGroup] eqn = LaplaceTransform[ y''[t] + 4y'[t] + 4y[t] == g[t],t,s] :[font = output; output; inactive; preserveAspect; endGroup] 4*LaplaceTransform[y[t], t, s] + s^2*LaplaceTransform[y[t], t, s] + 4*(s*LaplaceTransform[y[t], t, s] - y[0]) - s*y[0] - Derivative[1][y][0] == LaplaceTransform[g[t], t, s] ;[o] 4 LaplaceTransform[y[t], t, s] + 2 s LaplaceTransform[y[t], t, s] + 4 (s LaplaceTransform[y[t], t, s] - y[0]) - s y[0] - y'[0] =\ = LaplaceTransform[g[t], t, s] :[font = input; preserveAspect; startGroup] tran = Solve[eqn/.{y[0]->2, y'[0]->-3},Y[s]] :[font = output; output; inactive; preserveAspect; endGroup] {{LaplaceTransform[y[t], t, s] -> -((-5 - 2*s - LaplaceTransform[g[t], t, s])/(4 + 4*s + s^2)) }} ;[o] {{LaplaceTransform[y[t], t, s] -> -5 - 2 s - LaplaceTransform[g[t], t, s] -(---------------------------------------)}} 2 4 + 4 s + s :[font = text; inactive; preserveAspect] The inverse Laplace transform of the first two terms is straightforward: :[font = input; preserveAspect; startGroup] InverseLaplaceTransform[(5+2s)/(2+s)^2,s,t] :[font = output; output; inactive; preserveAspect; endGroup] 2/E^(2*t) + t/E^(2*t) ;[o] 2 t ---- + ---- 2 t 2 t E E :[font = text; inactive; preserveAspect] The last term is a product of F[s_] = 1/(2+s)^2 and G[s]. By the Convolution Theorem, the inverse Laplace transform of F[s] G[s] is the convolution integral of g[t]and ;[s] 11:0,0;29,1;47,0;52,1;56,0;119,1;123,0;124,1;128,0;160,1;164,0;169,-1; 2:6,13,9,Times,0,12,0,0,0;5,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] f[t_] = InverseLaplaceTransform[1/(2+s)^2,s,t] :[font = output; output; inactive; preserveAspect; endGroup] t/E^(2*t) ;[o] t ---- 2 t E :[font = input; preserveAspect; startGroup] Integrate[g[t-u]f[u],{u,0,t}] :[font = output; output; inactive; preserveAspect; endGroup] Integrate[(u*g[t - u])/E^(2*u), {u, 0, t}] ;[o] u g[t - u] Integrate[----------, {u, 0, t}] 2 u E :[font = text; inactive; preserveAspect] In one step: :[font = input; preserveAspect; startGroup] InverseLaplaceTransform[Y[s]/.First[tran],s,t] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 2/E^(2*t) + t/E^(2*t) + Integrate[(t10*g[t - t10])/E^(2*t10), {t10, 0, t}] ;[o] 2 t t10 g[t - t10] ---- + ---- + Integrate[--------------, {t10, 0, t}] 2 t 2 t 2 t10 E E E :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Problem 6 :[font = text; inactive; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450; endGroup] a) Show that the system of equations x1'[t] == a x1[t] + b x2[t] + f[t], x2'[t] == c x1[t] + d x2[t] + g[t], can be transformed into the second order equation y''[t] - (a + d) y[t] + (a d - b c) y[t] == f'[t] - d f[t] + b g[t] Note that a + d is the trace and a d - b c is the determinant of the 2x2 matrix {{a, b}, {c, d}}. (Hint: Solve the first equation for b x2[t] and substitute this into the second equation.) b) Solve the initial value problem x1'[t] == x1[t] - 2 x2[t], x2'[t] == 3 x1[t] - 4 x2[t], x1[0] == -1, x2[0] == 2. by transforming it into a second order equation as in a) and plot the solution for t>=0. ;[s] 26:0,2;3,0;37,1;116,0;167,1;247,0;258,1;263,0;271,2;276,0;282,1;291,0;299,2;312,0;330,1;346,0;384,1;391,0;439,2;441,0;474,1;569,0;623,2;625,0;652,1;656,0;658,-1; 3:13,13,9,Times,0,12,0,0,0;8,13,10,Courier,1,12,0,0,0;5,13,9,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] b x2[t] == x1'[t] - a x1[t] - f[t]; b x2'[t] == x1''[t] - a x1'[t] - f'[t]; :[font = text; inactive; preserveAspect] Substituting these into the second equation gives: :[font = input; preserveAspect] x1''[t] - a x1'[t] - f'[t] == b(c x1[t] + d x2[t] + g[t]) == b c x1[t] + d(x1'[t] - a x1[t] - f[t]) + b g[t]; :[font = text; inactive; preserveAspect] Let y[t] be x1[t]. Then the above equation simplifies to: ;[s] 5:0,0;4,1;8,0;12,1;17,0;58,-1; 2:3,13,9,Times,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect] y''[t] - a y'[t] - d y'[t] - b c y[t] + a d y[t] == f'[t] - d f[t] + b g[t]; :[font = text; inactive; preserveAspect] or: :[font = input; preserveAspect] y''[t] - (a + d) y[t] + (a d - b c) y[t] == f'[t] - d f[t] + b g[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] The matrix of coefficients in this case is :[font = input; preserveAspect] m = {{ 1, -2}, { 3, -4}}; :[font = input; preserveAspect; startGroup] Det[m] :[font = output; output; inactive; preserveAspect; endGroup] 2 ;[o] 2 :[font = text; inactive; preserveAspect] the trace is 1-4 = -3, and the functions f[t] and g[t] are both 0. In addition, y[t] == x1[t] so y[0] == x1[0] == -1, and x2[t] == (1/2)(x1[t] - x1'[t]) == (1/2)(y[t] - y'[t]) so 2 == x2[0] == (1/2)(y[0] - y'[0]) == (1/2)(-1-y'[0]) so y'[0] = -5. The second order equation associated to this system is thus ;[s] 19:0,0;13,1;21,0;41,1;45,0;50,1;54,0;64,1;65,0;80,1;93,0;97,1;116,0;122,1;175,0;179,1;231,0;235,1;245,0;307,-1; 2:10,13,9,Times,0,12,0,0,0;9,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect] y''[t] + 3 y'[t] + 2 y[t] == 0, y[0] == -1, y'[0] == -5 :[font = input; preserveAspect; startGroup] soln = DSolve[{y''[t] + 3 y'[t] + 2 y[t] == 0, y[0] == -1, y'[0] == -5},y[t],t] :[font = output; output; inactive; preserveAspect; endGroup] {{y[t] -> 6/E^(2*t) - 7/E^t}} ;[o] 6 7 {{y[t] -> ---- - --}} 2 t t E E :[font = input; preserveAspect; startGroup] y[t_] = y[t] /. First[soln] :[font = output; output; inactive; preserveAspect; endGroup] 6/E^(2*t) - 7/E^t ;[o] 6 7 ---- - -- 2 t t E E :[font = input; preserveAspect; startGroup] Plot[y[t],{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 0.603319 0.288303 [ [(2)] .21429 .60332 0 2 Msboxa [(4)] .40476 .60332 0 2 Msboxa [(6)] .59524 .60332 0 2 Msboxa [(8)] .78571 .60332 0 2 Msboxa [(10)] .97619 .60332 0 2 Msboxa [(-2)] .01131 .02671 1 0 Msboxa [(-1.5)] .01131 .17086 1 0 Msboxa [(-1)] .01131 .31502 1 0 Msboxa [(-0.5)] .01131 .45917 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 .21429 .60332 m .21429 .60957 L s P [(2)] .21429 .60332 0 2 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pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.97619 0.466485 0.25307 0.175124 [ [(-2)] .04322 .25307 0 2 Msboxa [(-1.5)] .27646 .25307 0 2 Msboxa [(-1)] .50971 .25307 0 2 Msboxa [(-0.5)] .74295 .25307 0 2 Msboxa [(-1)] .96369 .07795 1 0 Msboxa [(-0.5)] .96369 .16551 1 0 Msboxa [(0.5)] .96369 .34063 1 0 Msboxa [(1)] .96369 .42819 1 0 Msboxa [(1.5)] .96369 .51576 1 0 Msboxa [(2)] .96369 .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 .04322 .25307 m .04322 .25932 L s P [(-2)] .04322 .25307 0 2 Mshowa p .002 w .27646 .25307 m .27646 .25932 L s P [(-1.5)] .27646 .25307 0 2 Mshowa p .002 w .50971 .25307 m .50971 .25932 L s P [(-1)] .50971 .25307 0 2 Mshowa p .002 w .74295 .25307 m .74295 .25932 L s P [(-0.5)] .74295 .25307 0 2 Mshowa p .001 w .08987 .25307 m .08987 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matrix A = {{1,0,0,-1}, {0,-1,1,0}, {-1,0,1,0}, {0,1,-1,1}} b) Solve the system of linear equations x1 - x4 == 2, - x2 + x3 == 1, -x1 + x3 == -1, x2 - x3 + x4 == 0 by using Solve[], LinearSolve[], and by using the inverse of A from part a). ;[s] 16:0,2;2,0;37,1;121,2;123,0;161,1;281,0;290,1;297,0;299,1;312,0;342,1;343,0;354,2;356,0;357,1;363,-1; 3:7,13,9,Times,0,12,0,0,0;6,13,10,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) ;[s] 1:0,0;3,-1; 1:1,13,9,Times,2,12,0,0,0; :[font = input; preserveAspect] A = {{1,0,0,-1}, {0,-1,1,0}, {-1,0,1,0}, {0,1,-1,1}}; :[font = input; preserveAspect; startGroup] B = Inverse[A]; B//MatrixForm :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{1, 1, 0, 1}, {1, 0, 1, 1}, {1, 1, 1, 1}, {0, 1, 0, 1}}] ;[o] 1 1 0 1 1 0 1 1 1 1 1 1 0 1 0 1 :[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] Applying the inverse of A to the vector on the RHS of the equation gives: ;[s] 3:0,0;24,1;25,0;74,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] B.{2,1,-1,0} :[font = output; output; inactive; preserveAspect; endGroup] {3, 1, 2, 1} ;[o] {3, 1, 2, 1} :[font = text; inactive; preserveAspect] The solution is thus x1 == 3, x2 == 1, x3 == 2, x4 == 1. This output is similar to the result from LinearSolve[]: ;[s] 11:0,0;21,1;28,0;30,1;37,0;39,1;46,0;48,1;55,0;99,1;112,0;114,-1; 2:6,13,9,Times,0,12,0,0,0;5,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] ?LinearSolve :[font = print; inactive; preserveAspect; endGroup] LinearSolve[m, b] gives the vector x which solves the matrix equation m.x==b. :[font = input; preserveAspect; startGroup] LinearSolve[A,{2,1,-1,0}] :[font = output; output; inactive; preserveAspect; endGroup] {3, 1, 2, 1} ;[o] {3, 1, 2, 1} :[font = text; inactive; preserveAspect] Finally, using the general functions Solve[]: ;[s] 3:0,0;37,1;44,0;46,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] Solve[{x1 - x4 == 2, - x2 + x3 == 1, -x1 + x3 == -1, x2 - x3 + x4 == 0}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] {{x1 -> 3, x2 -> 1, x4 -> 1, x3 -> 2}} ;[o] {{x1 -> 3, x2 -> 1, x4 -> 1, x3 -> 2}} :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Problem 8 :[font = text; inactive; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450; endGroup] a) Find the eigenvalues of the matrix A = {{-4, 2, 11}, { 6, 1,-10}, {-5, 2, 12}} by solving Det[A - r*IdentityMatrix[3]] == 0. Then find the eigenvectors corresponding to each eigenvalue r by solving (A-r*IdentityMatrix[3]).{x1,x2,x3} == 0 Compare your results to the output of Eigensystem[]. b) Find a matrix T such that Inverse[T].A.T == d where d is a diagonal matrix (see p. 342). Verify your choice of T by calculating Inverse[T].A.T. ;[s] 24:0,2;2,0;39,1;105,0;116,1;149,0;211,1;212,0;224,1;268,0;306,1;319,0;321,2;323,0;338,1;339,0;350,1;369,0;376,1;377,0;435,1;436,0;452,1;466,0;468,-1; 3:12,13,9,Times,0,12,0,0,0;10,13,10,Courier,1,12,0,0,0;2,13,9,Times,2,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450] 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; startGroup] A = {{-4, 2, 11}, { 6, 1,-10}, {-5, 2, 12}}; Det[A - r*IdentityMatrix[3]] == 0 //Simplify :[font = output; output; inactive; preserveAspect; endGroup] 15 - 23*r + 9*r^2 - r^3 == 0 ;[o] 2 3 15 - 23 r + 9 r - r == 0 :[font = input; preserveAspect; startGroup] Solve[%] :[font = output; output; inactive; preserveAspect; endGroup] {{r -> 1}, {r -> 3}, {r -> 5}} ;[o] {{r -> 1}, {r -> 3}, {r -> 5}} :[font = text; inactive; preserveAspect] The eigenvalues are thus 1, 3 and 5. Now we find the corresponding eigenvectors. ;[s] 7:0,0;25,1;26,0;28,1;29,0;34,1;35,0;81,-1; 2:4,13,9,Times,0,12,0,0,0;3,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] eqns = (A-r*IdentityMatrix[3]).{x1,x2,x3} == {0,0,0} //Simplify :[font = output; output; inactive; preserveAspect; endGroup] {-4*x1 - r*x1 + 2*x2 + 11*x3, 6*x1 + x2 - r*x2 - 10*x3, -5*x1 + 2*x2 + 12*x3 - r*x3} == {0, 0, 0} ;[o] {-4 x1 - r x1 + 2 x2 + 11 x3, 6 x1 + x2 - r x2 - 10 x3, -5 x1 + 2 x2 + 12 x3 - r x3} == {0, 0, 0} :[font = text; inactive; preserveAspect] For the eigenvalue r == 1 we get ;[s] 3:0,0;19,1;25,0;33,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] Solve[eqns /. {r->1} ] :[font = output; output; inactive; preserveAspect; endGroup] {{x2 -> (-4*x3)/3, x1 -> (5*x3)/3}} ;[o] -4 x3 5 x3 {{x2 -> -----, x1 -> ----}} 3 3 :[font = text; inactive; preserveAspect] Choose x3 == 3, then x1 == 5, and x2 == -4, so {5,-4,3} is an eigenvector with eiugenvalue 1. Let's test it: ;[s] 11:0,0;7,1;14,0;21,1;28,0;34,1;42,0;47,1;55,0;91,1;92,0;109,-1; 2:6,13,9,Times,0,12,0,0,0;5,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] A.{5,-4,3} :[font = output; output; inactive; preserveAspect; endGroup] {5, -4, 3} ;[o] {5, -4, 3} :[font = text; inactive; preserveAspect] For the eigenvalue r == 3 we get ;[s] 3:0,0;19,1;25,0;33,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] Solve[eqns /. {r->3} ] :[font = output; output; inactive; preserveAspect; endGroup] {{x1 -> x3, x2 -> -2*x3}} ;[o] {{x1 -> x3, x2 -> -2 x3}} :[font = text; inactive; preserveAspect] Choose x3 == 1, then x1 == 1, and x2 == -2, so {1,-2,1} is an eigenvector with eiugenvalue 3. Let's test it: ;[s] 11:0,0;7,1;14,0;21,1;28,0;34,1;42,0;47,1;55,0;91,1;92,0;109,-1; 2:6,13,9,Times,0,12,0,0,0;5,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] A.{1,-2,1} :[font = output; output; inactive; preserveAspect; endGroup] {3, -6, 3} ;[o] {3, -6, 3} :[font = text; inactive; preserveAspect] For the eigenvalue r == 5 we get ;[s] 3:0,0;19,1;25,0;33,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] Solve[eqns /. {r->5} ] :[font = output; output; inactive; preserveAspect; endGroup] {{x1 -> x3, x2 -> -x3}} ;[o] {{x1 -> x3, x2 -> -x3}} :[font = text; inactive; preserveAspect] Choose x3 == 1, then x1 == 1, and x2 == -1, so {1,-1,1} is an eigenvector with eiugenvalue 5. Let's test it: ;[s] 11:0,0;7,1;14,0;21,1;28,0;34,1;42,0;47,1;55,0;91,1;92,0;109,-1; 2:6,13,9,Times,0,12,0,0,0;5,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] A.{1,-1,1} :[font = output; output; inactive; preserveAspect; endGroup] {5, -5, 5} ;[o] {5, -5, 5} :[font = text; inactive; preserveAspect] Now we try Eigensystem[]: ;[s] 3:0,0;11,1;24,0;26,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] ?Eigensystem :[font = print; inactive; preserveAspect; endGroup] Eigensystem[m] gives a list {values, vectors} of the eigenvalues and eigenvectors of the square matrix m. :[font = input; preserveAspect; startGroup] Eigensystem[A] :[font = output; output; inactive; preserveAspect; endGroup] {{1, 3, 5}, {{5, -4, 3}, {1, -2, 1}, {1, -1, 1}}} ;[o] {{1, 3, 5}, {{5, -4, 3}, {1, -2, 1}, {1, -1, 1}}} :[font = text; inactive; preserveAspect] Again we see that the eigenvalues are 1, 3, and 5, with corresponding eigenvectors {5, -4, 3}, {1, -2, 1}, and {1, -1, 1}. ;[s] 13:0,0;38,1;39,0;41,1;42,0;48,1;49,0;83,1;93,0;95,1;105,0;111,1;121,0;123,-1; 2:7,13,9,Times,0,12,0,0,0;6,13,10,Courier,1,12,0,0,0; :[font = text; inactive; preserveAspect] b) The matrix T should have columns given by the eigenvectors of A (see p.342): ;[s] 6:0,0;2,1;14,2;15,1;65,2;66,1;80,-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; startGroup] T = Transpose[{{5,-4,3},{1,-2,1},{1,-1,1}}]; T//MatrixForm :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{5, 1, 1}, {-4, -2, -1}, {3, 1, 1}}] ;[o] 5 1 1 -4 -2 -1 3 1 1 :[font = input; preserveAspect; startGroup] Inverse[T].A.T //Matrix :[font = output; output; inactive; preserveAspect; endGroup] {{1, 0, 0}, {0, 3, 0}, {0, 0, 5}} ;[o] {{1, 0, 0}, {0, 3, 0}, {0, 0, 5}} :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Problem 9 :[font = text; inactive; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450; endGroup] Consider the vectors x1 = {t, 1, t}; x2 = {1,t,t^2}; x3 = {t,t^2,t}; a) Compute the Wronskian of x1, x2, x3. b) At what points are the vectors x1, x2, x3 linearly independent? On what intervals are they linearly independent? c) What conclusion can be draw about the coefficients in the system of homogeneous differential equations satisfied by x1, x2, x3? d) Find such a system and verify the conclusions of part c). ;[s] 30:0,0;21,1;80,0;81,2;84,0;109,1;111,0;113,1;115,0;117,1;119,0;121,2;123,0;155,1;157,0;159,1;161,0;163,1;165,0;237,2;239,0;356,1;358,0;360,1;362,0;364,1;366,0;368,2;370,0;425,2;429,-1; 3:15,13,9,Times,0,12,0,0,0;10,13,10,Courier,1,12,0,0,0;5,13,9,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] x1 = {t, 1, t}; x2 = {1,t,t^2}; x3 = {t,t^2,t}; ;[s] 2:0,0;47,1;49,-1; 2:1,12,10,Courier,1,12,0,0,0;1,12,9,Times,0,12,0,0,0; :[font = input; preserveAspect; startGroup] w = Det[{x1,x2,x3}] :[font = output; output; inactive; preserveAspect; endGroup] -t + 2*t^3 - t^5 ;[o] 3 5 -t + 2 t - t :[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] Solve[w == 0] :[font = output; output; inactive; preserveAspect; endGroup] {{t -> -1}, {t -> -1}, {t -> 0}, {t -> 1}, {t -> 1}} ;[o] {{t -> -1}, {t -> -1}, {t -> 0}, {t -> 1}, {t -> 1}} :[font = text; inactive; preserveAspect] Since the Wronskian w is not zero at any point t not equal to -1, 0, or 1, the vectors x1, x2, and x3 are linearly independent at any point t not equal to -1, 0, or 1. We can see directly that they are dependent at the three points t == -1, 0, and 1: ;[s] 33:0,0;20,1;21,0;47,1;48,0;62,1;64,0;66,1;67,0;72,1;73,0;79,2;86,0;87,1;89,0;91,1;93,0;99,1;101,0;140,1;141,0;155,1;157,0;159,1;160,0;165,1;166,0;232,1;239,0;241,1;242,0;248,1;249,0;251,-1; 3:17,13,9,Times,0,12,0,0,0;15,13,10,Courier,1,12,0,0,0;1,13,9,Times,2,12,0,0,0; :[font = input; preserveAspect; startGroup] {x1,x2,x3} /. t->-1 //MatrixForm :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{-1, 1, -1}, {1, -1, 1}, {-1, 1, -1}}] ;[o] -1 1 -1 1 -1 1 -1 1 -1 :[font = text; inactive; preserveAspect] so, for example, 1*x1 + 1*x2 + 0*x3 == 0. ;[s] 2:0,0;17,1;42,-1; 2:1,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] {x1,x2,x3} /. t->0 //MatrixForm :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{0, 1, 0}, {1, 0, 0}, {0, 0, 0}}] ;[o] 0 1 0 1 0 0 0 0 0 :[font = text; inactive; preserveAspect] so 0*x1 + 0*x2 + 1*x3 == 0. ;[s] 2:0,0;3,1;28,-1; 2:1,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] {x1,x2,x3} /. t->1 //MatrixForm :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{1, 1, 1}, {1, 1, 1}, {1, 1, 1}}] ;[o] 1 1 1 1 1 1 1 1 1 :[font = text; inactive; preserveAspect] so, for example, 1*x1 - 1*x2 + 0*x3 == 0. ;[s] 2:0,0;17,1;42,-1; 2:1,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = text; inactive; preserveAspect] As vector functions, x1, x2, and x3 are still linearly independent on the entire real line.Vector functions being linearly dependent on an interval is equivalent to the Wronskian w of the vector functions being identically zero on that interval. Our Wronskian w is not 0 on any interval. ;[s] 17:0,0;3,2;19,0;21,1;23,0;25,1;27,0;33,1;35,0;179,1;180,0;260,1;261,0;269,1;270,0;278,2;286,0;288,-1; 3:9,13,9,Times,0,12,0,0,0;6,13,10,Courier,1,12,0,0,0;2,13,9,Times,2,12,0,0,0; :[font = text; inactive; preserveAspect] c) The coefficients in the system of homogeneous differential equations satisfied by x1, x2, x3 must be discontinuous at the point t == -1, 0, and 1. If they were continuous at these points, then Theorem 7.4.3 (p.348) would imply that the Wronskian w is never zero. (The assumption that the coefficients be continuous in Theorem 7.4.3 is actuall made on p. 345). ;[s] 16:0,1;2,0;86,2;88,0;90,2;92,0;94,2;96,0;132,2;139,0;141,2;142,0;148,2;149,0;250,2;251,0;364,-1; 3:8,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0;7,13,10,Courier,1,12,0,0,0; :[font = text; inactive; preserveAspect] d) To find a differential equation satisfied by x1, x2, and x3, let us solve for the coefficients aij in the following equations: ;[s] 10:0,0;2,1;49,2;51,1;53,2;55,1;61,2;63,1;99,2;102,1;131,-1; 3:1,13,9,Times,2,12,0,0,0;5,13,9,Times,0,12,0,0,0;4,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] eqns = {D[x1,t] == a11 x1 + a12 x2 + a13 x3, D[x2,t] == a21 x1 + a22 x2 + a23 x3, D[x3,t] == a31 x1 + a32 x2 + a33 x3} :[font = output; output; inactive; preserveAspect; endGroup] {{1, 0, 1} == {a12 + a11*t + a13*t, a11 + a12*t + a13*t^2, a11*t + a13*t + a12*t^2}, {0, 1, 2*t} == {a22 + a21*t + a23*t, a21 + a22*t + a23*t^2, a21*t + a23*t + a22*t^2}, {1, 2*t, 1} == {a32 + a31*t + a33*t, a31 + a32*t + a33*t^2, a31*t + a33*t + a32*t^2}} ;[o] 2 {{1, 0, 1} == {a12 + a11 t + a13 t, a11 + a12 t + a13 t , 2 a11 t + a13 t + a12 t }, 2 {0, 1, 2 t} == {a22 + a21 t + a23 t, a21 + a22 t + a23 t , 2 a21 t + a23 t + a22 t }, 2 {1, 2 t, 1} == {a32 + a31 t + a33 t, a31 + a32 t + a33 t , 2 a31 t + a33 t + a32 t }} :[font = input; preserveAspect; startGroup] Solve[eqns] :[font = message; inactive; preserveAspect] Solve::svars: Warning: Equations may not give solutions for all "solve" variables. :[font = output; output; inactive; preserveAspect; endGroup] {{a21 -> -(-1 + t^2)^(-1), a22 -> (2*t)/(-1 + t^2), a11 -> t/(-1 + t^2), a31 -> -(t/(-1 + t^2)), a23 -> -(-1 + t^2)^(-1), a12 -> 0, a32 -> 0, a13 -> -(1/(t*(-1 + t^2))), a33 -> (-1 + 2*t^2)/(t*(-1 + t^2))}} ;[o] 1 2 t t {{a21 -> -(-------), a22 -> -------, a11 -> -------, 2 2 2 -1 + t -1 + t -1 + t t 1 a31 -> -(-------), a23 -> -(-------), a12 -> 0, a32 -> 0, 2 2 -1 + t -1 + t 2 1 -1 + 2 t a13 -> -(-----------), a33 -> -----------}} 2 2 t (-1 + t ) t (-1 + t ) :[font = text; inactive; preserveAspect; endGroup] The coefficients, indeed, are not even all defined at the points t == -1, 0, and 1. ;[s] 7:0,0;66,1;73,0;75,1;76,0;82,1;83,0;85,-1; 2:4,13,9,Times,0,12,0,0,0;3,13,10,Courier,1,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Problem 10 :[font = text; inactive; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450; endGroup] Find the general solution of the system x1'[t] == x1[t] - 2x2[t], x2'[t] == 3x1[t] - 4x2[t] a) by determining the eigenvalues and eigenvectors of the matrix a = {{1,-2}, {3,-4}}; and using them to construct two linearly independent solutions of the form {z1,z2}E^(r*t). b) by using DSolve[]. c) Plot the trajectories for several of solutions found in a) using ParametricPlot[]. ;[s] 18:0,0;40,1;101,2;103,0;166,1;201,0;276,1;290,0;292,2;294,0;304,1;312,0;314,2;316,0;373,2;376,0;383,1;399,0;401,-1; 3:9,13,9,Times,0,12,0,0,0;5,13,10,Courier,1,12,0,0,0;4,13,9,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 = {{1,-2}, {3,-4}}; :[font = input; preserveAspect; startGroup] Eigensystem[a] :[font = output; output; inactive; preserveAspect; endGroup] {{-2, -1}, {{2, 3}, {1, 1}}} ;[o] {{-2, -1}, {{2, 3}, {1, 1}}} :[font = text; inactive; preserveAspect] This gives two fundamental solutions :[font = input; preserveAspect] X1[t_] = {1,1}*E^(-t); X2[t_] = {2,3}*E^(-2t); :[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*c2)/E^(2*t) + c1/E^t, (3*c2)/E^(2*t) + c1/E^t} ;[o] 2 c2 c1 3 c2 c1 {---- + --, ---- + --} 2 t t 2 t t E E E E :[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] DSolve[{x1'[t] == x1[t] - 2x2[t], x2'[t] == 3x1[t] - 4x2[t]},{x1[t],x2[t]},t] :[font = output; output; inactive; preserveAspect; endGroup] {{x1[t] -> (-2/E^(2*t) + 3/E^t)*C[1] + (2/E^(2*t) - 2/E^t)*C[2], x2[t] -> (-3/E^(2*t) + 3/E^t)*C[1] + (3/E^(2*t) - 2/E^t)*C[2]}} ;[o] -2 3 2 2 {{x1[t] -> (---- + --) C[1] + (---- - --) C[2], 2 t t 2 t t E E E E -3 3 3 2 x2[t] -> (---- + --) C[1] + (---- - --) C[2]}} 2 t t 2 t t E E E E :[font = text; inactive; preserveAspect] c) ;[s] 1:0,0;3,-1; 1:1,13,9,Times,2,12,0,0,0; :[font = text; inactive; preserveAspect] Let us plot the fundamental solutions, whose trajectories are lines, in different colors. 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.29309 m .50375 .29309 L s P p .001 w .5 .32495 m .50375 .32495 L s P p .001 w .5 .34088 m .50375 .34088 L s P p .001 w .5 .35681 m .50375 .35681 L s P p .001 w .5 .37274 m .50375 .37274 L s P p .001 w .5 .40461 m .50375 .40461 L s P p .001 w .5 .42054 m .50375 .42054 L s P p .001 w .5 .43647 m .50375 .43647 L s P p .001 w .5 .4524 m .50375 .4524 L s P p .001 w .5 .48427 m .50375 .48427 L s P p .001 w .5 .5002 m .50375 .5002 L s P p .001 w .5 .51613 m .50375 .51613 L s P p .001 w .5 .53206 m .50375 .53206 L s P p .001 w .5 .05411 m .50375 .05411 L s P p .001 w .5 .03818 m .50375 .03818 L s P p .001 w .5 .02225 m .50375 .02225 L s P p .001 w .5 .00631 m .50375 .00631 L s P p .001 w .5 .56393 m .50375 .56393 L s P p .001 w .5 .57986 m .50375 .57986 L s P p .001 w .5 .59579 m .50375 .59579 L s P p .001 w .5 .61172 m .50375 .61172 L s P p .002 w .5 0 m .5 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p 1 0 0 r p .004 w .97619 .60332 m .90309 .55814 L .84121 .51989 L .78882 .48752 L .74448 .46012 L .70695 .43692 L .67518 .41728 L .64829 .40066 L .62552 .38659 L .60625 .37468 L .58994 .3646 L .57613 .35607 L .56445 .34885 L .55455 .34273 L .54618 .33756 L .53909 .33317 L .53309 .32947 L .52801 .32633 L .52371 .32367 L .52007 .32142 L .51699 .31952 L .51438 .3179 L .51217 .31654 L .5103 .31538 L .50872 .31441 L s P P p 1 0 0 r p .004 w .02381 .01472 m .09691 .0599 L .15879 .09814 L .21118 .13051 L .25552 .15792 L .29305 .18111 L .32482 .20075 L .35171 .21737 L .37448 .23144 L .39375 .24335 L .41006 .25343 L .42387 .26196 L .43555 .26919 L .44545 .2753 L .45382 .28048 L .46091 .28486 L .46691 .28857 L .47199 .29171 L .47629 .29436 L .47993 .29661 L .48301 .29852 L .48562 .30013 L .48783 .30149 L .4897 .30265 L .49128 .30363 L s P P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; Cclosed; preserveAspect; startGroup] p2 = ParametricPlot[Evaluate[{X2[t],-X2[t]}],{t,-2,2}, PlotStyle->RGBColor[0,0,1]] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 358; pictureHeight = 221] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.0156903 0.309017 0.00646476 [ [(-30)] .02929 .30902 0 2 Msboxa [(-20)] .18619 .30902 0 2 Msboxa [(-10)] .3431 .30902 0 2 Msboxa [(10)] .6569 .30902 0 2 Msboxa [(20)] .81381 .30902 0 2 Msboxa [(30)] .97071 .30902 0 2 Msboxa [(-40)] .4875 .05043 1 0 Msboxa [(-20)] .4875 .17972 1 0 Msboxa [(20)] .4875 .43831 1 0 Msboxa [(40)] .4875 .56761 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 .02929 .30902 m .02929 .31527 L s P [(-30)] .02929 .30902 0 2 Mshowa p .002 w .18619 .30902 m .18619 .31527 L s P [(-20)] .18619 .30902 0 2 Mshowa p .002 w .3431 .30902 m .3431 .31527 L s P [(-10)] .3431 .30902 0 2 Mshowa p .002 w .6569 .30902 m .6569 .31527 L s P [(10)] .6569 .30902 0 2 Mshowa p .002 w .81381 .30902 m .81381 .31527 L s P [(20)] .81381 .30902 0 2 Mshowa p .002 w .97071 .30902 m .97071 .31527 L s P [(30)] .97071 .30902 0 2 Mshowa p .001 w .06067 .30902 m .06067 .31277 L s P p .001 w .09205 .30902 m .09205 .31277 L s P p .001 w .12343 .30902 m .12343 .31277 L s P p .001 w .15481 .30902 m .15481 .31277 L s P p .001 w .21757 .30902 m .21757 .31277 L s P p .001 w .24896 .30902 m .24896 .31277 L s P p .001 w .28034 .30902 m .28034 .31277 L s P p .001 w .31172 .30902 m .31172 .31277 L s P p .001 w .37448 .30902 m .37448 .31277 L s P p .001 w .40586 .30902 m .40586 .31277 L s P p .001 w .43724 .30902 m .43724 .31277 L s P p .001 w .46862 .30902 m .46862 .31277 L s P p .001 w .53138 .30902 m .53138 .31277 L s P p .001 w .56276 .30902 m .56276 .31277 L s P p .001 w .59414 .30902 m .59414 .31277 L s P p .001 w .62552 .30902 m .62552 .31277 L s P p .001 w .68828 .30902 m .68828 .31277 L s P p .001 w .71966 .30902 m .71966 .31277 L s P p .001 w .75104 .30902 m .75104 .31277 L s P p .001 w .78243 .30902 m .78243 .31277 L s P p .001 w .84519 .30902 m .84519 .31277 L s P p .001 w .87657 .30902 m .87657 .31277 L s P p .001 w .90795 .30902 m .90795 .31277 L s P p .001 w .93933 .30902 m .93933 .31277 L s P p .002 w 0 .30902 m 1 .30902 L s P p .002 w .5 .05043 m .50625 .05043 L s P [(-40)] .4875 .05043 1 0 Mshowa p .002 w .5 .17972 m .50625 .17972 L s P [(-20)] .4875 .17972 1 0 Mshowa p .002 w .5 .43831 m .50625 .43831 L s P [(20)] .4875 .43831 1 0 Mshowa p .002 w .5 .56761 m .50625 .56761 L s P [(40)] .4875 .56761 1 0 Mshowa p .001 w .5 .07629 m .50375 .07629 L s P p .001 w .5 .10214 m .50375 .10214 L s P p .001 w .5 .128 m .50375 .128 L s P p .001 w .5 .15386 m .50375 .15386 L s P p .001 w .5 .20558 m .50375 .20558 L s P p .001 w .5 .23144 m .50375 .23144 L s P p .001 w .5 .2573 m .50375 .2573 L s P p .001 w .5 .28316 m .50375 .28316 L s P p .001 w .5 .33488 m .50375 .33488 L s P p .001 w .5 .36074 m .50375 .36074 L s P p .001 w .5 .38659 m .50375 .38659 L s P p .001 w .5 .41245 m .50375 .41245 L s P p .001 w .5 .46417 m .50375 .46417 L s P p .001 w .5 .49003 m .50375 .49003 L s P p .001 w .5 .51589 m .50375 .51589 L s P p .001 w .5 .54175 m .50375 .54175 L s P p .001 w .5 .02457 m .50375 .02457 L s P p .001 w .5 .59347 m .50375 .59347 L s P p .002 w .5 0 m .5 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p 0 0 1 r p .004 w s s s s 1 .61803 m .95163 .58814 L s .95163 .58814 m .8236 .50902 L .73187 .45232 L .66614 .4117 L .61905 .38259 L .5853 .36174 L .56112 .34679 L .5438 .33608 L .53138 .32841 L .52249 .32291 L .51611 .31897 L .51154 .31615 L .50827 .31413 L .50593 .31268 L .50425 .31164 L .50304 .3109 L .50218 .31036 L .50156 .30998 L .50112 .30971 L .5008 .30951 L .50057 .30937 L s P P p 0 0 1 r p .004 w s s s s 0 0 m .04837 .0299 L s .04837 .0299 m .1764 .10902 L .26813 .16571 L .33386 .20633 L .38095 .23544 L .4147 .2563 L .43888 .27124 L .4562 .28195 L .46862 .28962 L .47751 .29512 L .48389 .29906 L .48846 .30188 L .49173 .3039 L .49407 .30535 L .49575 .30639 L .49696 .30714 L .49782 .30767 L .49844 .30805 L .49888 .30833 L .4992 .30852 L .49943 .30866 L s P P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = text; inactive; preserveAspect] We can use Table[] to generate a list of solutions for various choices of c1 and c2. We must use Flatten[...,1]to remove an extra layer of braces on the output of Table[]. Since the trajectories all approach the origin, we limit the plot range to a box around the origin. ;[s] 11:0,0;11,1;18,0;74,1;76,0;81,1;83,0;97,1;111,0;164,1;171,0;273,-1; 2:6,13,9,Times,0,12,0,0,0;5,13,10,Courier,1,12,0,0,0; :[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->{{-2,2},{-2,2}}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 279; pictureHeight = 172] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.25 0.309017 0.154508 [ [(-2)] 0 .30902 0 2 Msboxa [(-1.5)] .125 .30902 0 2 Msboxa [(-1)] .25 .30902 0 2 Msboxa [(-0.5)] .375 .30902 0 2 Msboxa [(0.5)] .625 .30902 0 2 Msboxa [(1)] .75 .30902 0 2 Msboxa [(1.5)] .875 .30902 0 2 Msboxa [(2)] 1 .30902 0 2 Msboxa [(-2)] .4875 0 1 0 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s s s s .47042 .61803 m .42716 .56494 L s .42716 .56494 m .31179 .40833 L .25 .30902 L .23321 .27492 L .22753 .26095 L .2234 .24878 L .22067 .23824 L .21977 .23353 L .21943 .2313 L .21916 .22917 L .21906 .22813 L .21896 .22711 L .21889 .22611 L .21883 .22513 L .21879 .22418 L .21876 .22324 L .21875 .22232 L .21875 .22142 L .21877 .22054 L .21881 .21967 L .21886 .21883 L .21892 .218 L .21909 .2164 L .21931 .21487 L .21959 .2134 L .21991 .212 L .22072 .20939 L .22171 .20703 L .22287 .20489 L .22419 .20297 L .22567 .20125 L .22729 .19973 L .22904 .1984 L .2329 .19624 L .23499 .19539 L .23718 .1947 L .23945 .19414 L .24062 .19391 L .24181 .19371 L .24301 .19354 L .24424 .19341 L .24548 .1933 L .24611 .19326 L .24674 .19322 L .24737 .19319 L .24801 .19317 L .24865 .19315 L .2493 .19314 L .24995 .19314 L .2506 .19314 L Mistroke .25125 .19315 L .25191 .19316 L .25324 .19321 L .25391 .19324 L .25458 .19328 L .25729 .19349 L .26004 .19378 L .26283 .19416 L .26849 .19512 L .28005 .19778 L .29176 .20123 L .31493 .20962 L .33704 .21904 L .35755 .22867 L .37618 .238 L .39287 .24676 L .40766 .25478 L Mfstroke P P p p .004 w s s s s s s s s s s s s s .61428 .61803 m .58167 .58091 L s .58167 .58091 m .47602 .45285 L .41298 .36892 L .37853 .3154 L .36886 .29692 L .36547 .28931 L .36293 .28267 L .36113 .27689 L .36048 .2743 L .35999 .27189 L .35965 .26967 L .35953 .26862 L .35949 .26811 L .35945 .26761 L .35942 .26712 L .3594 .26664 L .35938 .26617 L .35938 .26571 L .35938 .26526 L .35939 .26481 L .3594 .26438 L .35943 .26396 L .35946 .26354 L .35949 .26314 L .35959 .26235 L .35971 .2616 L .35986 .26089 L .36024 .25955 L .36071 .25833 L .36127 .25723 L .36191 .25624 L .36263 .25536 L .36342 .25457 L .36428 .25388 L .3652 .25327 L .36721 .25231 L .36829 .25194 L .36941 .25165 L .37058 .25142 L .37118 .25132 L .37179 .25125 L .3724 .25119 L .37271 .25116 L .37303 .25114 L .37334 .25112 L .37366 .2511 L .37398 .25109 L .3743 .25108 L .37462 .25108 L .37495 .25108 L Mistroke .37527 .25108 L .3756 .25108 L .37593 .25109 L .37626 .2511 L .37693 .25113 L .3776 .25117 L .37828 .25122 L .37965 .25136 L .38244 .25175 L .38529 .25228 L .38818 .25293 L .39402 .25454 L .40565 .25861 L .41681 .26327 L Mfstroke P P p p .004 w s s s s s s s s s s s s s s s .66742 .61803 m .59692 .53944 L s .59692 .53944 m .51033 .43759 L .45736 .37022 L .42709 .32669 L .41799 .31143 L .41191 .29952 L .40825 .29036 L .40765 .28844 L .40717 .28665 L .4068 .28499 L .40665 .28421 L .40653 .28346 L .40643 .28273 L .40635 .28203 L .4063 .28136 L .40628 .28103 L .40626 .28072 L .40625 .2804 L .40625 .2801 L .40625 .2798 L .40626 .2795 L .40627 .27922 L .40629 .27894 L .40631 .27866 L .40633 .27839 L .4064 .27787 L .40648 .27737 L .4067 .27644 L .40684 .27601 L .40699 .27559 L .40733 .27482 L .40774 .27413 L .40819 .27351 L .40925 .27246 L .40984 .27203 L .41048 .27166 L .41115 .27134 L .41185 .27108 L .41259 .27086 L .41336 .27068 L .41415 .27055 L .41455 .2705 L .41497 .27046 L .41538 .27043 L .41559 .27042 L .4158 .27041 L .41602 .2704 L .41623 .27039 L .41645 .27039 L .41666 .27039 L Mistroke .41688 .27039 L .4171 .27039 L .41732 .2704 L .41754 .27041 L .41798 .27043 L .41843 .27045 L .41934 .27053 L .42026 .27064 L .42213 .27092 L .42597 .27175 L Mfstroke P P p p .004 w s s s s s s s s s s s s s s s s s s .10452 0 m .11306 .00714 L s .11306 .00714 m .19884 .07794 L .26398 .13094 L .31375 .17083 L .35205 .20104 L .38172 .22407 L .40486 .24173 L s P P p p .004 w s s s s s s s s s s s s s s s s s s .09429 0 m .18072 .06987 L s .18072 .06987 m .24733 .12289 L .29872 .16315 L .33865 .19391 L .36989 .21758 L .3945 .23591 L .41402 .25022 L s P P p p .004 w s s s s s s s s s s s s s s s s .07543 0 m .11149 .02818 L s .11149 .02818 m .18826 .08717 L .24839 .1326 L .29581 .16784 L .33346 .19535 L .36354 .21699 L .38773 .23412 L .40728 .24776 L .42317 .25871 L s P P p p .004 w s s s s s s s s s s s s 0 0 m 0 0 L s 0 0 m .07676 .04744 L .14173 .0876 L .19673 .12159 L .24329 .15036 L .2827 .17472 L .31606 .19534 L .3443 .21279 L .3682 .22756 L .38843 .24007 L .40556 .25065 L .42006 .25961 L .43233 .2672 L s P P p p .004 w s s s s s s s s s s s .63132 .61803 m .60713 .59086 L s .60713 .59086 m .5 .46353 L .43502 .37957 L .39844 .32558 L .38768 .30675 L .38067 .29211 L .37834 .28613 L .37668 .28092 L .37608 .27858 L .37562 .27641 L .37544 .27539 L .37529 .2744 L .37518 .27346 L .37513 .273 L .37509 .27255 L .37506 .2721 L .37503 .27167 L .37501 .27125 L .375 .27083 L .375 .27042 L .375 .27002 L .37501 .26963 L .37503 .26925 L .37508 .26851 L .37512 .26815 L .37516 .2678 L .37538 .26647 L .37553 .26585 L .3757 .26526 L .3761 .26416 L .37658 .26317 L .37714 .26227 L .37846 .26075 L .37921 .26012 L .38002 .25956 L .38087 .25909 L .38178 .25868 L .38273 .25834 L .38373 .25807 L .38476 .25785 L .38529 .25776 L .38582 .25769 L .38637 .25763 L .38692 .25758 L .3872 .25756 L .38748 .25755 L .38776 .25754 L .38805 .25753 L .38833 .25752 L .38862 .25752 L Mistroke .38891 .25751 L .3892 .25752 L .38949 .25752 L .38978 .25753 L .39037 .25755 L .39067 .25756 L .39097 .25758 L .39157 .25762 L .39278 .25774 L .39402 .25789 L .39526 .25807 L .39779 .25853 L .40294 .25977 L .41333 .26314 L .4234 .26719 L .43284 .27146 L .44149 .27569 L Mfstroke P P p p .004 w s s s s s s s s s s s s s s .70595 .61803 m .65515 .56356 L s .65515 .56356 m .56461 .46263 L .50689 .39473 L .47158 .34982 L .4514 .3208 L .44533 .31063 L .44127 .30269 L .43883 .29658 L .43844 .2953 L .43812 .29411 L .43787 .293 L .43777 .29248 L .43768 .29198 L .43762 .29149 L .43757 .29103 L .43753 .29058 L .43752 .29036 L .43751 .29015 L .4375 .28994 L .4375 .28974 L .4375 .28954 L .43751 .28934 L .43751 .28915 L .43752 .28896 L .43754 .28878 L .43756 .2886 L .4376 .28825 L .43766 .28792 L .4378 .2873 L .43789 .28701 L .43799 .28673 L .43822 .28622 L .43849 .28576 L .43879 .28534 L .4395 .28465 L .4399 .28436 L .44032 .28411 L .44077 .2839 L .44124 .28372 L .44173 .28358 L .44224 .28346 L .44277 .28337 L .44304 .28334 L .44331 .28331 L .44359 .28329 L .44373 .28328 L .44387 .28328 L .44401 .28327 L .44415 .28327 L .4443 .28327 L Mistroke .44444 .28327 L .44459 .28327 L .44473 .28327 L .44488 .28327 L .44503 .28328 L .44532 .28329 L .44562 .28331 L .44623 .28336 L .44684 .28343 L .44809 .28362 L .45065 .28418 L Mfstroke P P p p .004 w s s s s s s s s s s s s s s s s .73426 .61803 m .63869 .51691 L s .63869 .51691 m .56601 .43736 L .51906 .38353 L .48976 .34764 L .47243 .32418 L .46312 .3093 L .46057 .30418 L .45971 .30209 L .45907 .30026 L .45884 .29944 L .45865 .29867 L .45851 .29797 L .45845 .29763 L .45841 .29731 L .45838 .297 L .45836 .29685 L .45835 .2967 L .45834 .29656 L .45834 .29642 L .45833 .29628 L .45833 .29614 L .45833 .29601 L .45834 .29588 L .45834 .29575 L .45835 .29563 L .45837 .29539 L .45839 .29527 L .4584 .29516 L .45849 .29472 L .45854 .29452 L .4586 .29433 L .45874 .29397 L .45891 .29365 L .4591 .29336 L .45931 .2931 L .45981 .29267 L s P P p p .004 w s s s s s s s s s s s s s s s s s s .13181 0 m .20503 .06398 L s .20503 .06398 m .27669 .12606 L .32988 .17167 L .36954 .20531 L .39927 .23023 L .42169 .24877 L .43869 .26264 L s P P p p .004 w s s s s s s s s s s s s s s s s s .12507 0 m .20247 .06677 L s .20247 .06677 m .27269 .12671 L .32518 .171 L .36462 .20387 L .39443 .22839 L .41711 .24676 L .43447 .26062 L .44785 .27113 L s P P p p .004 w s s s s s s s s s s s s s s s .11126 0 m .16443 .04478 L s .16443 .04478 m .23985 .10751 L .29691 .15432 L .34036 .18945 L .37366 .21595 L .39936 .23608 L .41932 .25146 L .43494 .2633 L .44725 .27247 L .45701 .27962 L s P P p p .004 w s s s s s s s s 0 0 m .01307 .00808 L s .01307 .00808 m .08782 .05428 L .1511 .09338 L .20466 .12649 L .25 .15451 L .28838 .17823 L .32087 .19831 L .34837 .2153 L .37165 .22969 L .39135 .24187 L .40803 .25218 L .42215 .2609 L .4341 .26829 L .44422 .27454 L .45278 .27983 L .46003 .28431 L .46617 .28811 L s P P p p .004 w s s s s s s s s s s s s .75 .61803 m .75 .61803 L s .75 .61803 m .64665 .51036 L .57758 .43629 L .53231 .38582 L .50344 .35187 L .48579 .32942 L .4757 .31491 L .47266 .30982 L .47064 .30585 L .46942 .3028 L .46922 .30216 L .46906 .30156 L .46893 .30101 L .46888 .30075 L .46884 .3005 L .46881 .30025 L .46878 .30002 L .46877 .2998 L .46876 .29969 L .46875 .29958 L .46875 .29948 L .46875 .29938 L .46875 .29928 L .46875 .29918 L .46876 .29908 L .46876 .29899 L .46877 .2989 L .46878 .29881 L .4688 .29863 L .46883 .29847 L .4689 .29816 L .46895 .29801 L .469 .29788 L .46911 .29762 L .46925 .29739 L .4694 .29718 L .46975 .29683 L .46995 .29669 L .47016 .29657 L .47038 .29646 L .47062 .29637 L .47086 .2963 L .47112 .29624 L .47138 .2962 L .47152 .29618 L .47166 .29616 L .47179 .29615 L .47186 .29615 L .47193 .29615 L .47201 .29614 L Mistroke .47208 .29614 L .47215 .29614 L .47222 .29614 L .47229 .29614 L .47237 .29614 L .47244 .29614 L .47251 .29615 L .47266 .29615 L .47281 .29616 L .47311 .29619 L .47342 .29622 L .47404 .29632 L .47532 .2966 L Mfstroke P P p p .004 w s s s s s s s s s s s s s s s .77799 .61803 m .71625 .55635 L s .71625 .55635 m .63524 .47406 L .58023 .41697 L .54337 .37764 L .51912 .3508 L .50358 .3327 L .494 .3207 L .48846 .31291 L .48675 .31016 L .48559 .30801 L .48518 .30712 L .48486 .30635 L .48448 .30509 L s P P p p .004 w s s s s s s s s s s s s s s s s .78925 .61803 m .76704 .59624 L s .76704 .59624 m .67466 .50451 L .61103 .44037 L .56761 .39575 L .53833 .36491 L .5189 .34377 L .50629 .32944 L .49837 .31986 L .49364 .31358 L s P P p p .004 w s s s s s s s s s s s s s s s s s .16667 0 m .21669 .04637 L s .21669 .04637 m .297 .12082 L .35454 .17417 L .39577 .21239 L .42532 .23978 L .44649 .25941 L .46166 .27347 L .47253 .28355 L s P P p p .004 w s s s s s s s s s s s s s s s s .16667 0 m .2364 .06465 L s .2364 .06465 m .31112 .13392 L .36466 .18355 L .40303 .21912 L .43052 .2446 L .45021 .26286 L .46433 .27595 L .47444 .28532 L .48168 .29204 L s P P p p .004 w s s s s s s s s s s s s s s .16667 0 m .24329 .07104 L s .24329 .07104 m .31606 .1385 L .3682 .18683 L .40556 .22147 L .43233 .24629 L .45151 .26407 L .46526 .27681 L .47511 .28594 L .48216 .29248 L .48722 .29717 L .49084 .30053 L s P P p p .004 w .5 .30902 m .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L s P P p p .004 w s s s s s s s s s s s s s s .83333 .61803 m .75671 .547 L s .75671 .547 m .68394 .47954 L .6318 .4312 L .59444 .39657 L .56767 .37175 L .54849 .35397 L .53474 .34122 L .52489 .33209 L .51784 .32555 L .51278 .32087 L .50916 .31751 L s P P p p .004 w s s s s s s s s s s s s s s s s .83333 .61803 m .7636 .55338 L s .7636 .55338 m .68888 .48411 L .63534 .43448 L .59697 .39891 L .56948 .37343 L .54979 .35517 L .53567 .34209 L .52556 .33271 L .51832 .326 L s P P p p .004 w s s s s s s s s s s s s s s s s s .83333 .61803 m .78331 .57166 L s .78331 .57166 m .703 .49721 L .64546 .44386 L .60423 .40564 L .57468 .37825 L .55351 .35862 L .53834 .34456 L .52747 .33449 L s P P p p .004 w s s s s s s s s s s s s s s s s .21075 0 m .23296 .02179 L s .23296 .02179 m .32534 .11352 L .38897 .17766 L .43239 .22228 L .46167 .25312 L .4811 .27426 L .49371 .28859 L .50163 .29817 L .50636 .30446 L s P P p p .004 w s s s s s s s s s s s s s s s .22201 0 m .28375 .06169 L s .28375 .06169 m .36476 .14398 L .41977 .20107 L .45663 .24039 L .48088 .26723 L .49642 .28533 L .506 .29734 L .51154 .30513 L .51325 .30787 L .51441 .31002 L .51482 .31091 L .51514 .31169 L .51552 .31295 L s P P p p .004 w s s s s s s s s s s s s .25 0 m .25 0 L s .25 0 m .35335 .10768 L .42242 .18175 L .46769 .23221 L .49656 .26616 L .51421 .28862 L .5243 .30313 L .52734 .30821 L .52936 .31218 L .53058 .31524 L .53078 .31588 L .53094 .31647 L .53107 .31702 L .53112 .31729 L .53116 .31754 L .53119 .31778 L .53122 .31801 L .53123 .31824 L .53124 .31834 L .53125 .31845 L .53125 .31855 L .53125 .31866 L .53125 .31876 L .53125 .31885 L .53124 .31895 L .53124 .31904 L .53123 .31914 L .53122 .31923 L .5312 .3194 L .53117 .31957 L .5311 .31988 L .53105 .32002 L .531 .32016 L .53089 .32041 L .53075 .32065 L .5306 .32085 L .53025 .3212 L .53005 .32134 L .52984 .32147 L .52962 .32157 L .52938 .32166 L .52914 .32174 L .52888 .3218 L .52862 .32184 L .52848 .32186 L .52834 .32187 L .52821 .32188 L .52814 .32188 L .52807 .32189 L .52799 .32189 L Mistroke .52792 .32189 L .52785 .32189 L .52778 .32189 L .52771 .32189 L .52763 .32189 L .52756 .32189 L .52749 .32189 L .52734 .32188 L .52719 .32187 L .52689 .32185 L .52658 .32181 L .52596 .32172 L .52468 .32144 L Mfstroke P P p p .004 w s s s s s s s s 1 .61803 m .98693 .60996 L s .98693 .60996 m .91218 .56376 L .8489 .52465 L .79534 .49155 L .75 .46353 L .71162 .43981 L .67913 .41973 L .65163 .40273 L .62835 .38834 L .60865 .37617 L .59197 .36586 L .57785 .35713 L .5659 .34974 L .55578 .34349 L .54722 .3382 L .53997 .33372 L .53383 .32993 L s P P p p .004 w s s s s s s s s s s s s s s s .88874 .61803 m .83557 .57325 L s .83557 .57325 m .76015 .51053 L .70309 .46371 L .65964 .42859 L .62634 .40208 L .60064 .38195 L .58068 .36657 L .56506 .35474 L .55275 .34557 L .54299 .33842 L s P P p p .004 w s s s s s s s s s s s s s s s s s .87493 .61803 m .79753 .55126 L s .79753 .55126 m .72731 .49132 L .67482 .44703 L .63538 .41416 L .60557 .38965 L .58289 .37127 L .56553 .35742 L .55215 .34691 L s P P p p .004 w s s s s s s s s s s s s s s s s s s .86819 .61803 m .79497 .55405 L s .79497 .55405 m .72331 .49198 L .67012 .44637 L .63046 .41273 L .60073 .38781 L .57831 .36927 L .56131 .3554 L s P P p p .004 w s s s s s s s s s s s s s s s s .26574 0 m .36131 .10112 L s .36131 .10112 m .43399 .18067 L .48094 .2345 L .51024 .2704 L .52757 .29385 L .53688 .30874 L .53943 .31385 L .54029 .31595 L .54093 .31778 L .54116 .3186 L .54135 .31936 L .54149 .32007 L .54155 .3204 L .54159 .32072 L .54162 .32103 L .54164 .32118 L .54165 .32133 L .54166 .32148 L .54166 .32162 L .54167 .32175 L .54167 .32189 L .54167 .32202 L .54166 .32215 L .54166 .32228 L .54165 .32241 L .54163 .32265 L .54161 .32276 L .5416 .32288 L .54151 .32331 L .54146 .32351 L .5414 .3237 L .54126 .32406 L .54109 .32438 L .5409 .32467 L .54069 .32493 L .54019 .32537 L s P P p p .004 w s s s s s s s s s s s s s s .29405 0 m .34485 .05447 L s .34485 .05447 m .43539 .1554 L .49311 .2233 L .52842 .26822 L .5486 .29724 L .55467 .30741 L .55873 .31535 L .56117 .32145 L .56156 .32274 L .56188 .32393 L .56213 .32503 L .56223 .32556 L .56232 .32606 L .56238 .32654 L .56243 .32701 L .56247 .32745 L .56248 .32767 L .56249 .32788 L .5625 .32809 L .5625 .3283 L .5625 .3285 L .56249 .32869 L .56249 .32888 L .56248 .32907 L .56246 .32925 L .56244 .32943 L .5624 .32978 L .56234 .33011 L .5622 .33074 L .56211 .33102 L .56201 .3313 L .56178 .33181 L .56151 .33227 L .56121 .33269 L .5605 .33339 L .5601 .33367 L .55968 .33392 L .55923 .33413 L .55876 .33431 L .55827 .33446 L .55776 .33457 L .55723 .33466 L .55696 .33469 L .55669 .33472 L .55641 .33474 L .55627 .33475 L .55613 .33476 L .55599 .33476 L .55585 .33477 L .5557 .33477 L Mistroke .55556 .33477 L .55541 .33477 L .55527 .33477 L .55512 .33476 L .55497 .33476 L .55468 .33474 L .55438 .33473 L .55377 .33467 L .55316 .3346 L .55191 .33441 L .54935 .33386 L Mfstroke P P p p .004 w s s s s s s s s s s s .36868 0 m .39287 .02718 L s .39287 .02718 m .5 .15451 L .56498 .23846 L .60156 .29246 L .61232 .31129 L .61933 .32592 L .62166 .33191 L .62332 .33712 L .62392 .33945 L .62438 .34162 L .62456 .34264 L .62471 .34363 L .62482 .34458 L .62487 .34504 L .62491 .34549 L .62494 .34593 L .62497 .34636 L .62499 .34679 L .625 .3472 L .625 .34761 L .625 .34801 L .62499 .3484 L .62497 .34878 L .62492 .34952 L .62488 .34988 L .62484 .35023 L .62462 .35156 L .62447 .35218 L .6243 .35277 L .6239 .35387 L .62342 .35487 L .62286 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.45884 .29944 L .45865 .29867 L .45851 .29797 L .45845 .29763 L .45841 .29731 L .45838 .297 L .45836 .29685 L .45835 .2967 L .45834 .29656 L .45834 .29642 L .45833 .29628 L .45833 .29614 L .45833 .29601 L .45834 .29588 L .45834 .29575 L .45835 .29563 L .45837 .29539 L .45839 .29527 L .4584 .29516 L .45849 .29472 L .45854 .29452 L .4586 .29433 L .45874 .29397 L .45891 .29365 L .4591 .29336 L .45931 .2931 L .45981 .29267 L s P P p p .004 w s s s s s s s s s s s s s s s s s s .13181 0 m .20503 .06398 L s .20503 .06398 m .27669 .12606 L .32988 .17167 L .36954 .20531 L .39927 .23023 L .42169 .24877 L .43869 .26264 L s P P p p .004 w s s s s s s s s s s s s s s s s s .12507 0 m .20247 .06677 L s .20247 .06677 m .27269 .12671 L .32518 .171 L .36462 .20387 L .39443 .22839 L .41711 .24676 L .43447 .26062 L .44785 .27113 L s P P p p .004 w s s s s s s s s s s s s s s s .11126 0 m .16443 .04478 L s .16443 .04478 m .23985 .10751 L .29691 .15432 L .34036 .18945 L .37366 .21595 L .39936 .23608 L .41932 .25146 L .43494 .2633 L .44725 .27247 L .45701 .27962 L s P P p p .004 w s s s s s s s s 0 0 m .01307 .00808 L s .01307 .00808 m .08782 .05428 L .1511 .09338 L .20466 .12649 L .25 .15451 L .28838 .17823 L .32087 .19831 L .34837 .2153 L .37165 .22969 L .39135 .24187 L .40803 .25218 L .42215 .2609 L .4341 .26829 L .44422 .27454 L .45278 .27983 L .46003 .28431 L .46617 .28811 L s P P p p .004 w s s s s s s s s s s s s .75 .61803 m .75 .61803 L s .75 .61803 m .64665 .51036 L .57758 .43629 L .53231 .38582 L .50344 .35187 L .48579 .32942 L .4757 .31491 L .47266 .30982 L .47064 .30585 L .46942 .3028 L .46922 .30216 L .46906 .30156 L .46893 .30101 L .46888 .30075 L .46884 .3005 L .46881 .30025 L .46878 .30002 L .46877 .2998 L .46876 .29969 L .46875 .29958 L .46875 .29948 L .46875 .29938 L .46875 .29928 L .46875 .29918 L .46876 .29908 L .46876 .29899 L .46877 .2989 L .46878 .29881 L .4688 .29863 L .46883 .29847 L .4689 .29816 L .46895 .29801 L .469 .29788 L .46911 .29762 L .46925 .29739 L .4694 .29718 L .46975 .29683 L .46995 .29669 L .47016 .29657 L .47038 .29646 L .47062 .29637 L .47086 .2963 L .47112 .29624 L .47138 .2962 L .47152 .29618 L .47166 .29616 L .47179 .29615 L .47186 .29615 L .47193 .29615 L .47201 .29614 L Mistroke .47208 .29614 L .47215 .29614 L .47222 .29614 L .47229 .29614 L .47237 .29614 L .47244 .29614 L .47251 .29615 L .47266 .29615 L .47281 .29616 L .47311 .29619 L .47342 .29622 L .47404 .29632 L .47532 .2966 L Mfstroke P P p p .004 w s s s s s s s s s s s s s s s .77799 .61803 m .71625 .55635 L s .71625 .55635 m .63524 .47406 L .58023 .41697 L .54337 .37764 L .51912 .3508 L .50358 .3327 L .494 .3207 L .48846 .31291 L .48675 .31016 L .48559 .30801 L .48518 .30712 L .48486 .30635 L .48448 .30509 L s P P p p .004 w s s s s s s s s s s s s s s s s .78925 .61803 m .76704 .59624 L s .76704 .59624 m .67466 .50451 L .61103 .44037 L .56761 .39575 L .53833 .36491 L .5189 .34377 L .50629 .32944 L .49837 .31986 L .49364 .31358 L s P P p p .004 w s s s s s s s s s s s s s s s s s .16667 0 m .21669 .04637 L s .21669 .04637 m .297 .12082 L .35454 .17417 L .39577 .21239 L .42532 .23978 L .44649 .25941 L .46166 .27347 L .47253 .28355 L s P P p p .004 w s s s s s s s s s s s s s s s s .16667 0 m .2364 .06465 L s .2364 .06465 m .31112 .13392 L .36466 .18355 L .40303 .21912 L .43052 .2446 L .45021 .26286 L .46433 .27595 L .47444 .28532 L .48168 .29204 L s P P p p .004 w s s s s s s s s s s s s s s .16667 0 m .24329 .07104 L s .24329 .07104 m .31606 .1385 L .3682 .18683 L .40556 .22147 L .43233 .24629 L .45151 .26407 L .46526 .27681 L .47511 .28594 L .48216 .29248 L .48722 .29717 L .49084 .30053 L s P P p p .004 w .5 .30902 m .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L .5 .30902 L s P P p p .004 w s s s s s s s s s s s s s s .83333 .61803 m .75671 .547 L s .75671 .547 m .68394 .47954 L .6318 .4312 L .59444 .39657 L .56767 .37175 L .54849 .35397 L .53474 .34122 L .52489 .33209 L .51784 .32555 L .51278 .32087 L .50916 .31751 L s P P p p .004 w s s s s s s s s s s s s s s s s .83333 .61803 m .7636 .55338 L s .7636 .55338 m .68888 .48411 L .63534 .43448 L .59697 .39891 L .56948 .37343 L .54979 .35517 L .53567 .34209 L .52556 .33271 L .51832 .326 L s P P p p .004 w s s s s s s s s s s s s s s s s s .83333 .61803 m .78331 .57166 L s .78331 .57166 m .703 .49721 L .64546 .44386 L .60423 .40564 L .57468 .37825 L .55351 .35862 L .53834 .34456 L .52747 .33449 L s P P p p .004 w s s s s s s s s s s s s s s s s .21075 0 m .23296 .02179 L s .23296 .02179 m .32534 .11352 L .38897 .17766 L .43239 .22228 L .46167 .25312 L .4811 .27426 L .49371 .28859 L .50163 .29817 L .50636 .30446 L s P P p p .004 w s s s s s s s s s s s s s s s .22201 0 m .28375 .06169 L s .28375 .06169 m .36476 .14398 L .41977 .20107 L .45663 .24039 L .48088 .26723 L .49642 .28533 L .506 .29734 L .51154 .30513 L .51325 .30787 L .51441 .31002 L .51482 .31091 L .51514 .31169 L .51552 .31295 L s P P p p .004 w s s s s s s s s s s s s .25 0 m .25 0 L s .25 0 m .35335 .10768 L .42242 .18175 L .46769 .23221 L .49656 .26616 L .51421 .28862 L .5243 .30313 L .52734 .30821 L .52936 .31218 L .53058 .31524 L .53078 .31588 L .53094 .31647 L .53107 .31702 L .53112 .31729 L .53116 .31754 L .53119 .31778 L .53122 .31801 L .53123 .31824 L .53124 .31834 L .53125 .31845 L .53125 .31855 L .53125 .31866 L .53125 .31876 L .53125 .31885 L .53124 .31895 L .53124 .31904 L .53123 .31914 L .53122 .31923 L .5312 .3194 L .53117 .31957 L .5311 .31988 L .53105 .32002 L .531 .32016 L .53089 .32041 L .53075 .32065 L .5306 .32085 L .53025 .3212 L .53005 .32134 L .52984 .32147 L .52962 .32157 L .52938 .32166 L .52914 .32174 L .52888 .3218 L .52862 .32184 L .52848 .32186 L .52834 .32187 L .52821 .32188 L .52814 .32188 L .52807 .32189 L .52799 .32189 L Mistroke .52792 .32189 L .52785 .32189 L .52778 .32189 L .52771 .32189 L .52763 .32189 L .52756 .32189 L .52749 .32189 L .52734 .32188 L .52719 .32187 L .52689 .32185 L .52658 .32181 L .52596 .32172 L .52468 .32144 L Mfstroke P P p p .004 w s s s s s s s s 1 .61803 m .98693 .60996 L s .98693 .60996 m .91218 .56376 L .8489 .52465 L .79534 .49155 L .75 .46353 L .71162 .43981 L .67913 .41973 L .65163 .40273 L .62835 .38834 L .60865 .37617 L .59197 .36586 L .57785 .35713 L .5659 .34974 L .55578 .34349 L .54722 .3382 L .53997 .33372 L .53383 .32993 L s P P p p .004 w s s s s s s s s s s s s s s s .88874 .61803 m .83557 .57325 L s .83557 .57325 m .76015 .51053 L .70309 .46371 L .65964 .42859 L .62634 .40208 L .60064 .38195 L .58068 .36657 L .56506 .35474 L .55275 .34557 L .54299 .33842 L s P P p p .004 w s s s s s s s s s s s s s s s s s .87493 .61803 m .79753 .55126 L s .79753 .55126 m .72731 .49132 L .67482 .44703 L .63538 .41416 L .60557 .38965 L .58289 .37127 L .56553 .35742 L .55215 .34691 L s P P p p .004 w s s s s s s s s s s s s s s s s s s .86819 .61803 m .79497 .55405 L s .79497 .55405 m .72331 .49198 L .67012 .44637 L .63046 .41273 L .60073 .38781 L .57831 .36927 L .56131 .3554 L s P P p p .004 w s s s s s s s s s s s s s s s s .26574 0 m .36131 .10112 L s .36131 .10112 m .43399 .18067 L .48094 .2345 L .51024 .2704 L .52757 .29385 L .53688 .30874 L .53943 .31385 L .54029 .31595 L .54093 .31778 L .54116 .3186 L .54135 .31936 L .54149 .32007 L .54155 .3204 L .54159 .32072 L .54162 .32103 L .54164 .32118 L .54165 .32133 L .54166 .32148 L .54166 .32162 L .54167 .32175 L .54167 .32189 L .54167 .32202 L .54166 .32215 L .54166 .32228 L .54165 .32241 L .54163 .32265 L .54161 .32276 L .5416 .32288 L .54151 .32331 L .54146 .32351 L .5414 .3237 L .54126 .32406 L .54109 .32438 L .5409 .32467 L .54069 .32493 L .54019 .32537 L s P P p p .004 w s s s s s s s s s s s s s s .29405 0 m .34485 .05447 L s .34485 .05447 m .43539 .1554 L .49311 .2233 L .52842 .26822 L .5486 .29724 L .55467 .30741 L .55873 .31535 L .56117 .32145 L .56156 .32274 L .56188 .32393 L .56213 .32503 L .56223 .32556 L .56232 .32606 L .56238 .32654 L .56243 .32701 L .56247 .32745 L .56248 .32767 L .56249 .32788 L .5625 .32809 L .5625 .3283 L .5625 .3285 L .56249 .32869 L .56249 .32888 L .56248 .32907 L .56246 .32925 L .56244 .32943 L .5624 .32978 L .56234 .33011 L .5622 .33074 L .56211 .33102 L .56201 .3313 L .56178 .33181 L .56151 .33227 L .56121 .33269 L .5605 .33339 L .5601 .33367 L .55968 .33392 L .55923 .33413 L .55876 .33431 L .55827 .33446 L .55776 .33457 L .55723 .33466 L .55696 .33469 L .55669 .33472 L .55641 .33474 L .55627 .33475 L .55613 .33476 L .55599 .33476 L .55585 .33477 L .5557 .33477 L Mistroke .55556 .33477 L .55541 .33477 L .55527 .33477 L .55512 .33476 L .55497 .33476 L .55468 .33474 L .55438 .33473 L .55377 .33467 L .55316 .3346 L .55191 .33441 L .54935 .33386 L Mfstroke P P p p .004 w s s s s s s s s s s s .36868 0 m .39287 .02718 L s .39287 .02718 m .5 .15451 L .56498 .23846 L .60156 .29246 L .61232 .31129 L .61933 .32592 L .62166 .33191 L .62332 .33712 L .62392 .33945 L .62438 .34162 L .62456 .34264 L .62471 .34363 L .62482 .34458 L .62487 .34504 L .62491 .34549 L .62494 .34593 L .62497 .34636 L .62499 .34679 L .625 .3472 L .625 .34761 L .625 .34801 L .62499 .3484 L .62497 .34878 L .62492 .34952 L .62488 .34988 L .62484 .35023 L .62462 .35156 L .62447 .35218 L .6243 .35277 L .6239 .35387 L .62342 .35487 L .62286 .35577 L .62154 .35729 L .62079 .35792 L .61998 .35847 L .61913 .35895 L .61822 .35935 L .61727 .35969 L .61627 .35997 L .61524 .36018 L .61471 .36027 L .61418 .36034 L .61363 .3604 L .61308 .36045 L .6128 .36047 L .61252 .36048 L .61224 .3605 L .61195 .36051 L .61167 .36051 L .61138 .36052 L Mistroke .61109 .36052 L .6108 .36052 L .61051 .36051 L .61022 .36051 L .60963 .36048 L .60933 .36047 L .60903 .36045 L .60843 .36041 L .60722 .3603 L .60598 .36015 L .60474 .35996 L .60221 .3595 L .59706 .35827 L .58667 .35489 L .5766 .35085 L .56716 .34657 L .55851 .34235 L Mfstroke P P p p .004 w s s s s s s s s s s s s 1 .61803 m 1 .61803 L s 1 .61803 m .92324 .57059 L .85827 .53044 L .80327 .49645 L .75671 .46767 L .7173 .44332 L .68394 .4227 L .6557 .40525 L .6318 .39047 L .61157 .37797 L .59444 .36738 L .57994 .35842 L .56767 .35084 L s P P p p .004 w s s s s s s s s s s s s s s s s .92457 .61803 m .88851 .58986 L s .88851 .58986 m .81174 .53086 L .75161 .48543 L .70419 .45019 L .66654 .42268 L .63646 .40105 L .61227 .38392 L .59272 .37027 L .57683 .35933 L s P P p p .004 w s s s s s s s s s s s s s s s s s s .90571 .61803 m .81928 .54816 L s .81928 .54816 m .75267 .49514 L .70128 .45489 L .66135 .42412 L .63011 .40045 L .6055 .38212 L .58598 .36782 L s P P p p .004 w s s s s s s s s s s s s s s s s s s .89548 .61803 m .88694 .61089 L s .88694 .61089 m .80116 .54009 L .73602 .4871 L .68625 .4472 L .64795 .41699 L .61828 .39397 L .59514 .37631 L s P P p p .004 w s s s s s s s s s s s s s s s .33258 0 m .40308 .07859 L s .40308 .07859 m .48967 .18045 L .54264 .24782 L .57291 .29134 L .58201 .3066 L .58809 .31851 L .59175 .32767 L .59235 .32959 L .59283 .33138 L .5932 .33304 L .59335 .33382 L .59347 .33458 L .59357 .3353 L .59365 .336 L .5937 .33667 L .59372 .337 L .59374 .33732 L .59375 .33763 L .59375 .33794 L .59375 .33824 L .59374 .33853 L .59373 .33882 L .59371 .3391 L .59369 .33937 L .59367 .33964 L .5936 .34016 L .59352 .34066 L .5933 .34159 L .59316 .34203 L .59301 .34244 L .59267 .34321 L .59226 .3439 L .59181 .34453 L .59075 .34557 L .59016 .346 L .58952 .34637 L .58885 .34669 L .58815 .34696 L .58741 .34718 L .58664 .34735 L .58585 .34748 L .58545 .34753 L .58503 .34757 L .58462 .3476 L .58441 .34762 L .5842 .34763 L .58398 .34763 L .58377 .34764 L .58355 .34764 L .58334 .34764 L Mistroke .58312 .34764 L .5829 .34764 L .58268 .34763 L .58246 .34763 L .58202 .34761 L .58157 .34758 L .58066 .3475 L .57974 .3474 L .57787 .34711 L .57403 .34628 L Mfstroke P P p p .004 w s s s s s s s s s s s s s .38572 0 m .41833 .03712 L s .41833 .03712 m .52398 .16518 L .58702 .24912 L .62147 .30263 L .63114 .32112 L .63453 .32872 L .63707 .33537 L .63887 .34115 L .63952 .34374 L .64001 .34614 L .64035 .34837 L .64047 .34942 L .64051 .34993 L .64055 .35043 L .64058 .35092 L .6406 .3514 L .64062 .35187 L .64062 .35233 L .64062 .35278 L .64061 .35322 L .6406 .35365 L .64057 .35408 L .64054 .35449 L .64051 .3549 L .64041 .35568 L .64029 .35643 L .64014 .35715 L .63976 .35849 L .63929 .3597 L .63873 .3608 L .63809 .36179 L .63737 .36268 L .63658 .36346 L .63572 .36416 L .6348 .36476 L .63279 .36572 L .63171 .36609 L .63059 .36639 L .62942 .36662 L .62882 .36671 L .62821 .36679 L .6276 .36685 L .62729 .36687 L .62697 .3669 L .62666 .36691 L .62634 .36693 L .62602 .36694 L .6257 .36695 L .62538 .36696 L .62505 .36696 L Mistroke .62473 .36696 L .6244 .36695 L .62407 .36695 L .62374 .36694 L .62307 .36691 L .6224 .36687 L .62172 .36681 L .62035 .36668 L .61756 .36629 L .61471 .36576 L .61182 .3651 L .60598 .36349 L .59435 .35943 L .58319 .35477 L Mfstroke P P p p .004 w s s s s s s s s s s .52958 0 m .57284 .05309 L s .57284 .05309 m .68821 .20971 L .75 .30902 L .76679 .34312 L .77247 .35708 L .7766 .36925 L .77933 .3798 L .78023 .38451 L .78057 .38673 L .78084 .38887 L .78094 .38991 L .78104 .39092 L .78111 .39192 L .78117 .3929 L .78121 .39386 L .78124 .3948 L .78125 .39571 L .78125 .39662 L .78123 .3975 L .78119 .39836 L .78114 .39921 L .78108 .40003 L .78091 .40163 L .78069 .40317 L .78041 .40463 L .78009 .40603 L .77928 .40864 L .77829 .41101 L .77713 .41314 L .77581 .41507 L .77433 .41678 L .77271 .4183 L .77096 .41964 L .7671 .4218 L .76501 .42264 L .76282 .42334 L .76055 .42389 L .75938 .42412 L .75819 .42432 L .75699 .42449 L .75576 .42463 L .75452 .42473 L .75389 .42478 L .75326 .42481 L .75263 .42484 L .75199 .42487 L .75135 .42488 L .7507 .42489 L .75005 .4249 L .7494 .4249 L Mistroke .74875 .42489 L .74809 .42487 L .74676 .42482 L .74609 .42479 L .74542 .42475 L .74271 .42454 L .73996 .42425 L .73717 .42388 L .73151 .42292 L .71995 .42025 L .70824 .41681 L .68507 .40841 L .66296 .39899 L .64245 .38937 L .62382 .38003 L .60713 .37128 L .59234 .36326 L Mfstroke P P p p .004 w s s s s s s s s s s s s s s s 1 .61803 m .9549 .59016 L s .9549 .59016 m .88506 .547 L .82595 .51046 L .77591 .47954 L .73355 .45336 L .6977 .4312 L .66735 .41244 L .64166 .39657 L .61991 .38313 L .6015 .37175 L s P P p p .004 w s s s s s s s s s s s s s s s s s .94753 .61803 m .92039 .59801 L s .92039 .59801 m .84358 .54227 L .78204 .49831 L .73244 .46341 L .69224 .43552 L .65949 .4131 L .63269 .39497 L .61066 .38024 L s P P p p .004 w s s s s s s s s s s s s s s s s s s .928 .61803 m .91124 .605 L s .91124 .605 m .83052 .54326 L .76718 .49562 L .71713 .4586 L .67733 .42964 L .64547 .40682 L .61982 .38873 L s P P p p .004 w s s s s s s s s s s s s s s s s s s s .91648 .61803 m .87901 .58821 L s .87901 .58821 m .80192 .52782 L .74203 .48168 L .69517 .44617 L .65825 .41867 L .62897 .39722 L s P P P p p 1 0 0 r p .004 w s s s s s s s s 1 .61803 m .98693 .60996 L s .98693 .60996 m .91218 .56376 L .8489 .52465 L .79534 .49155 L .75 .46353 L .71162 .43981 L .67913 .41973 L .65163 .40273 L .62835 .38834 L .60865 .37617 L .59197 .36586 L .57785 .35713 L .5659 .34974 L .55578 .34349 L .54722 .3382 L .53997 .33372 L .53383 .32993 L s P P p 1 0 0 r p .004 w s s s s s s s s 0 0 m .01307 .00808 L s .01307 .00808 m .08782 .05428 L .1511 .09338 L .20466 .12649 L .25 .15451 L .28838 .17823 L .32087 .19831 L .34837 .2153 L .37165 .22969 L .39135 .24187 L .40803 .25218 L .42215 .2609 L .4341 .26829 L .44422 .27454 L .45278 .27983 L .46003 .28431 L .46617 .28811 L s P P P p p 0 0 1 r p .004 w s s s s s s s s s s s s s s .83333 .61803 m .75671 .547 L s .75671 .547 m .68394 .47954 L .6318 .4312 L .59444 .39657 L .56767 .37175 L .54849 .35397 L .53474 .34122 L .52489 .33209 L .51784 .32555 L .51278 .32087 L .50916 .31751 L s P P p 0 0 1 r p .004 w s s s s s s s s s s s s s s .16667 0 m .24329 .07104 L s .24329 .07104 m .31606 .1385 L .3682 .18683 L .40556 .22147 L .43233 .24629 L .45151 .26407 L .46526 .27681 L .47511 .28594 L .48216 .29248 L .48722 .29717 L .49084 .30053 L s P P P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = text; inactive; preserveAspect] The plot shows the trajectories approach the origin tangent to the trajectory of X1[t](the red line). ;[s] 3:0,0;81,1;86,0;102,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; ^*)