(*^ ::[ 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. The line below identifies what version of Mathematica created this file, but it can be opened using any other version as well."; FrontEndVersion = "Macintosh Mathematica Notebook Front End Version 2.2"; MacintoshStandardFontEncoding; fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e8, 24, "Times"; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e6, 18, "Times"; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, e6, 14, "Times"; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, a20, 18, "Times"; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, a15, 14, "Times"; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, 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"Times"; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; paletteColors = 128; showRuler; currentKernel; ] :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] Math 325: Differential Equations Solutions to Assignment 1 :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Initialization :[font = input; initialization; noKeepOnOnePage; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450; endGroup] *) <True] :[font = output; output; inactive; preserveAspect; endGroup] 1 ;[o] 1 :[font = text; inactive; preserveAspect; endGroup] Since w is never 0, the functions are linearly independent on any interval. ;[s] 5:0,0;6,1;7,0;17,1;18,0;76,-1; 2:3,13,9,Times,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Problem 2 :[font = text; inactive; preserveAspect] a) Show that Sin[3x] is linearly dependent on the functions Sin[x] and Sin[x]^3 by finding constants c1, c2, c3 such that Sin[3x] = c1 Sin[x] + c2 Sin[x]^3. b) Show that the Wronskian of Sin[x], Sin[x]^3, and Sin[3x] is identically 0. ;[s] 25:0,2;4,0;14,1;21,0;61,1;67,0;72,1;80,0;102,1;104,0;106,1;108,0;110,1;112,0;123,1;156,0;158,2;162,0;189,1;195,0;197,1;205,0;211,1;218,0;234,1;237,-1; 3:12,13,9,Times,0,12,0,0,0;11,13,10,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) Use the trig identities for sums: Sin[a+b] == Sin[a]Cos[b] + Sin[b]Cos[a]; Cos[a+b] == Cos[a]Cos[b] - Sin[a]Sin[b]; ;[s] 3:0,2;4,0;38,1;128,-1; 3:1,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0;1,13,9,Times,2,12,0,0,0; :[font = input; preserveAspect; startGroup] Sin[3x] == Sin[2x]Cos[x] + Sin[x]Cos[2x] == 2Sin[x]Cos[x]^2 + Sin[x](Cos[x]^2-Sin[x]^2) == 2Sin[x](1-Sin[x]^2) + Sin[x](1-2Sin[x]^2) == 3Sin[x] - 4Sin[x]^3 :[font = text; inactive; preserveAspect] The two plots are identical: :[font = input; preserveAspect; startGroup] Plot[{Sin[3x],3Sin[x]-4Sin[x]^3},{x,0,2Pi}] :[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 0.151576 0.309017 0.294302 [ [(1)] .17539 .30902 0 2 Msboxa [(2)] .32696 .30902 0 2 Msboxa [(3)] .47854 .30902 0 2 Msboxa [(4)] .63011 .30902 0 2 Msboxa [(5)] .78169 .30902 0 2 Msboxa [(6)] .93327 .30902 0 2 Msboxa [(-1)] .01131 .01472 1 0 Msboxa [(-0.5)] .01131 .16187 1 0 Msboxa [(0.5)] .01131 .45617 1 0 Msboxa [(1)] .01131 .60332 1 0 Msboxa [ -0.001 -0.001 0 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.93651 .10091 L .95635 .19639 L .97619 .30902 L Mfstroke P P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = text; inactive; preserveAspect] b) The Wronskian is ;[s] 2:0,0;2,1;22,-1; 2:1,13,9,Times,2,12,0,0,0;1,13,9,Times,0,12,0,0,0; :[font = input; preserveAspect; startGroup] m = {{ Sin[3x], Sin[x], Sin[x]^3}, { D[Sin[3x],x], D[Sin[x],x], D[Sin[x]^3,x] }, { D[Sin[3x],{x,2}],D[Sin[x],{x,2}],D[Sin[x]^3,{x,2}]}} :[font = output; output; inactive; preserveAspect; endGroup] {{Sin[3*x], Sin[x], Sin[x]^3}, {3*Cos[3*x], Cos[x], 3*Cos[x]*Sin[x]^2}, {-9*Sin[3*x], -Sin[x], 6*Cos[x]^2*Sin[x] - 3*Sin[x]^3}} ;[o] 3 {{Sin[3 x], Sin[x], Sin[x] }, 2 {3 Cos[3 x], Cos[x], 3 Cos[x] Sin[x] }, 2 3 {-9 Sin[3 x], -Sin[x], 6 Cos[x] Sin[x] - 3 Sin[x] }} :[font = input; preserveAspect; startGroup] w = Det[m] :[font = output; output; inactive; preserveAspect; endGroup] -18*Cos[x]^2*Cos[3*x]*Sin[x]^2 + 6*Cos[3*x]*Sin[x]^4 + 6*Cos[x]^3*Sin[x]*Sin[3*x] - 18*Cos[x]*Sin[x]^3*Sin[3*x] ;[o] 2 2 4 -18 Cos[x] Cos[3 x] Sin[x] + 6 Cos[3 x] Sin[x] + 3 3 6 Cos[x] Sin[x] Sin[3 x] - 18 Cos[x] Sin[x] Sin[3 x] :[font = input; preserveAspect; startGroup] Simplify[%] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 0 ;[o] 0 :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Problem 3 :[font = text; inactive; preserveAspect] Find all the fifth roots of 2, i.e. find all complex numbers z = r Cos[t] + I r Sin[t] such that z^5 == 2. Plot these points {r Cos[t], r Sin[t]} using ListPlot[]. ;[s] 11:0,0;28,1;29,0;61,1;86,0;97,1;105,0;125,1;145,0;152,1;162,0;164,-1; 2:6,13,9,Times,0,12,0,0,0;5,13,10,Courier,1,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450] Solution :[font = text; inactive; preserveAspect] In polar form, z = r E^(I*t), so if z^5 == 2, then r^5 == 2, and E^(5*I*t) == 1. Thus, r = 2^(1/5) and 5*t = 2Pi*k where k==0, 1, 2, 3, or 4. The fifth roots of 2 in rectangular form are thus: ;[s] 27:0,0;15,1;28,0;36,1;44,0;51,1;59,0;65,1;79,0;87,1;98,0;103,1;114,0;121,1;125,0;127,1;128,0;130,1;131,0;133,1;134,0;139,1;140,0;141,1;142,0;161,1;162,0;193,-1; 2:14,13,9,Times,0,12,0,0,0;13,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] Table[{2^(1/5)Cos[2Pi*k/5],2^(1/5)Sin[2Pi*k/5]}, {k,0,4}] :[font = output; output; inactive; preserveAspect; endGroup] {{2^(1/5), 0}, {2^(1/5)*Cos[(2*Pi)/5], 2^(1/5)*Sin[(2*Pi)/5]}, {2^(1/5)*Cos[(4*Pi)/5], 2^(1/5)*Sin[(4*Pi)/5]}, {2^(1/5)*Cos[(6*Pi)/5], 2^(1/5)*Sin[(6*Pi)/5]}, {2^(1/5)*Cos[(8*Pi)/5], 2^(1/5)*Sin[(8*Pi)/5]}} ;[o] 1/5 1/5 2 Pi 1/5 2 Pi {{2 , 0}, {2 Cos[----], 2 Sin[----]}, 5 5 1/5 4 Pi 1/5 4 Pi {2 Cos[----], 2 Sin[----]}, 5 5 1/5 6 Pi 1/5 6 Pi {2 Cos[----], 2 Sin[----]}, 5 5 1/5 8 Pi 1/5 8 Pi {2 Cos[----], 2 Sin[----]}} 5 5 :[font = input; preserveAspect; startGroup] N[%] :[font = output; output; inactive; preserveAspect; endGroup] {{1.148698354997035, 0}, {0.3549673131046301, 1.092477055777454}, {-0.929316490603148, 0.6751879523998812}, {-0.929316490603148, -0.6751879523998812}, {0.3549673131046301, -1.092477055777454}} ;[o] {{1.1487, 0}, {0.354967, 1.09248}, {-0.929316, 0.675188}, {-0.929316, -0.675188}, {0.354967, -1.09248}} :[font = input; preserveAspect; startGroup] ListPlot[%, 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 0.449727 0.458313 0.5 0.435881 [ [(-0.5)] .22057 .5 0 2 Msboxa [(0.5)] .67888 .5 0 2 Msboxa [(1)] .90804 .5 0 2 Msboxa [(-1)] .43723 .06412 1 0 Msboxa [(-0.5)] .43723 .28206 1 0 Msboxa [(0.5)] .43723 .71794 1 0 Msboxa [(1)] .43723 .93588 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 1.001 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .22057 .5 m .22057 .50625 L s P [(-0.5)] .22057 .5 0 2 Mshowa p .002 w .67888 .5 m .67888 .50625 L s P [(0.5)] .67888 .5 0 2 Mshowa p .002 w .90804 .5 m .90804 .50625 L s P [(1)] .90804 .5 0 2 Mshowa p .001 w .03725 .5 m .03725 .50375 L s P p .001 w .08308 .5 m .08308 .50375 L s P p .001 w .12891 .5 m .12891 .50375 L s P p .001 w .17474 .5 m .17474 .50375 L s P p .001 w .2664 .5 m .2664 .50375 L s P p .001 w .31223 .5 m .31223 .50375 L s P p .001 w .35806 .5 m .35806 .50375 L s P p .001 w .4039 .5 m .4039 .50375 L s P p .001 w .49556 .5 m .49556 .50375 L s P p .001 w .54139 .5 m .54139 .50375 L s P p .001 w .58722 .5 m .58722 .50375 L s P p .001 w .63305 .5 m .63305 .50375 L s P p .001 w .72471 .5 m .72471 .50375 L s P p .001 w .77055 .5 m .77055 .50375 L s P p .001 w .81638 .5 m .81638 .50375 L s P p .001 w .86221 .5 m .86221 .50375 L s P p .001 w .95387 .5 m .95387 .50375 L s P p .001 w .9997 .5 m .9997 .50375 L s P p .002 w 0 .5 m 1 .5 L s P p .002 w .44973 .06412 m .45598 .06412 L s P [(-1)] .43723 .06412 1 0 Mshowa p .002 w .44973 .28206 m .45598 .28206 L s P [(-0.5)] .43723 .28206 1 0 Mshowa p .002 w .44973 .71794 m .45598 .71794 L s P [(0.5)] .43723 .71794 1 0 Mshowa p .002 w .44973 .93588 m .45598 .93588 L s P [(1)] .43723 .93588 1 0 Mshowa p .001 w .44973 .10771 m .45348 .10771 L s P p .001 w .44973 .15129 m .45348 .15129 L s P p .001 w .44973 .19488 m .45348 .19488 L s P p .001 w .44973 .23847 m .45348 .23847 L s P p .001 w .44973 .32565 m .45348 .32565 L s P p .001 w .44973 .36924 m .45348 .36924 L s P p .001 w .44973 .41282 m .45348 .41282 L s P p .001 w .44973 .45641 m .45348 .45641 L s P p .001 w .44973 .54359 m .45348 .54359 L s P p .001 w .44973 .58718 m .45348 .58718 L s P p .001 w .44973 .63076 m .45348 .63076 L s P p .001 w .44973 .67435 m .45348 .67435 L s P p .001 w .44973 .76153 m .45348 .76153 L s P p .001 w .44973 .80512 m .45348 .80512 L s P p .001 w .44973 .84871 m .45348 .84871 L s P p .001 w .44973 .89229 m .45348 .89229 L s P p .001 w .44973 .02053 m .45348 .02053 L s P p .001 w .44973 .97947 m .45348 .97947 L s P p .002 w .44973 0 m .44973 1 L s P P 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath p .008 w .97619 .5 Mdot .61241 .97619 Mdot .02381 .7943 Mdot .02381 .2057 Mdot .61241 .02381 Mdot 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 4 :[font = text; inactive; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] Find the solution of the differential equation y''''[x] - 8 y'[x] == 0 by 1) solving the characteristic equation to find fundamental solutions; 2) using DSolve[]. Superimpose plots of several solutions for various choices of the constants. ;[s] 11:0,0;47,1;74,0;78,1;82,0;152,1;156,0;165,1;173,0;174,1;175,0;252,-1; 2:6,13,9,Times,0,12,0,0,0;5,13,10,Courier,1,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450] Solution :[font = text; inactive; preserveAspect] The characteristic equation is r^4 - 8r == 0 which factors as r(r^3-2^3) == r(r-2)(r^2+2r+4) == 0 ;[s] 4:0,0;31,1;44,0;62,1;98,-1; 2:2,13,9,Times,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] Solve[r^4 - 8r == 0] :[font = output; output; inactive; preserveAspect; endGroup] {{r -> 0}, {r -> 2}, {r -> (-2 - 2*I*3^(1/2))/2}, {r -> (-2 + 2*I*3^(1/2))/2}} ;[o] -2 - 2 I Sqrt[3] {{r -> 0}, {r -> 2}, {r -> ----------------}, 2 -2 + 2 I Sqrt[3] {r -> ----------------}} 2 :[font = text; inactive; preserveAspect] The corresponding fundamental solutions are 1, E^(2x), E^(-x)*Cos[Sqrt[3]x], and E^(-x)*Sin[Sqrt[3]x] and the general soution to the homogeneous equation is ;[s] 9:0,0;44,1;45,0;47,1;53,0;55,1;75,0;81,1;101,0;157,-1; 2:5,13,9,Times,0,12,0,0,0;4,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect] phi[x_,c1_,c2_,c3_,c4_] = c1 + c2*E^(2x) + E^(-x)(c3*Cos[Sqrt[3]x]+c4*Sin[Sqrt[3]x]); :[font = input; preserveAspect; startGroup] phi[x,1,1,1,1] :[font = output; output; inactive; preserveAspect; endGroup] 1 + E^(2*x) + (Cos[3^(1/2)*x] + Sin[3^(1/2)*x])/E^x ;[o] 2 x Cos[Sqrt[3] x] + Sin[Sqrt[3] x] 1 + E + ------------------------------- x E :[font = input; preserveAspect] Table[phi[x,c1,c2,c3,c4],{c1,-1,1},{c2,-1,1},{c3,-1,1}, {c4,-1,1}] :[font = input; preserveAspect; startGroup] Plot[Evaluate[Table[phi[x,c1,c2,c3,c4], {c1,0,1},{c2,0,1},{c3,0,1},{c4,0,1}]], {x,0,Pi}, PlotRange->{-1,4}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica 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.06349 .35288 L .10317 .33274 L .14286 .3121 L .18254 .29233 L .22222 .27441 L .2619 .25899 L .30159 .24641 L .34127 .23678 L .36111 .23304 L .38095 .22999 L .40079 .22759 L .41071 .22662 L .42063 .2258 L .43056 .22512 L .44048 .22457 L .4504 .22415 L .45536 .22399 L .46032 .22386 L .46528 .22375 L .46776 .22371 L .47024 .22367 L .47272 .22365 L .4752 .22363 L .47644 .22362 L .47768 .22361 L .47892 .22361 L .48016 .2236 L .4814 .2236 L .48264 .2236 L .48388 .2236 L .48512 .22361 L .48636 .22361 L .4876 .22362 L .49008 .22363 L .49256 .22366 L .49504 .22369 L .5 .22376 L .50496 .22386 L .50992 .22398 L .51984 .22428 L .53968 .22509 L .55952 .22617 L .57937 .22745 L .61905 .23044 L .65873 .23375 L .69841 .23711 L .7381 .2403 L .77778 .24317 L .81746 .24562 L Mistroke .85714 .2476 L .87698 .24841 L .89683 .2491 L .91667 .24968 L .93651 .25014 L .95635 .25051 L .97619 .25077 L Mfstroke P P p p .004 w .02381 .37082 m .03373 .37343 L .04365 .37534 L .04861 .37605 L .05357 .37661 L .05605 .37683 L .05853 .37701 L .06101 .37715 L .06349 .37726 L .06473 .37729 L .06597 .37733 L .06721 .37735 L .06845 .37736 L .06969 .37736 L .07093 .37736 L .07217 .37735 L .07341 .37732 L .07589 .37725 L .07713 .37721 L .07837 .37715 L .08333 .37685 L .08829 .37642 L .09325 .37587 L .10317 .37441 L .1131 .37252 L .12302 .37023 L .14286 .3646 L .18254 .35 L .22222 .33261 L .2619 .3141 L .30159 .29584 L .34127 .27888 L .38095 .26392 L .42063 .25141 L .46032 .24152 L .48016 .23757 L .5 .23426 L .51984 .23156 L .53968 .22944 L .55952 .22787 L .56944 .22728 L .57937 .2268 L .58929 .22644 L .59425 .22631 L .59921 .22619 L .60417 .22611 L .60665 .22607 L .60913 .22605 L .61161 .22602 L .61409 .22601 L Mistroke .61533 .226 L .61657 .226 L .61781 .226 L .61905 .22599 L .62029 .226 L .62153 .226 L .62277 .226 L .62401 .226 L .62649 .22602 L .62897 .22604 L .63145 .22606 L .63393 .22609 L .63889 .22616 L .64881 .22637 L .65873 .22664 L .67857 .2274 L .69841 .22838 L .7381 .23084 L .77778 .23372 L 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2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] DSolve[y''''[x] - 8 y'[x] == 0,y[x],x] :[font = output; output; inactive; preserveAspect; endGroup] {{y[x] -> C[1] + E^(2*x)*C[2] + C[3]/E^(2*(-1)^(1/3)*x) + E^(2*(-1)^(2/3)*x)*C[4]}} ;[o] 2/3 2 x C[3] 2 (-1) x {{y[x] -> C[1] + E C[2] + ------------ + E C[4]}} 1/3 2 (-1) x E :[font = text; inactive; preserveAspect; endGroup] This is actually the same answer, except that Mathematica has kept the answer in complex number form ((-1)^(1/3) == (-1 + Sqrt[3])/2). ;[s] 5:0,0;46,1;57,0;102,2;132,0;135,-1; 3:3,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Problem 5 :[font = text; inactive; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] Find the solution of the initial value problem y'''[x] - y''[x] + y'[x] - y[x] == 0, y[Pi/2] == 2, y'[Pi/2] == 1, y''[Pi/2] == 0 by 1) solving the characteristic equation to find fundamental solutions of the homogeneous equationand using the initial conditions to solve for the constants; 2) using DSolve[]. Plot the solution. ;[s] 13:0,0;47,1;136,0;140,1;144,0;219,1;226,0;307,1;311,0;320,1;328,0;329,1;330,0;349,-1; 2:7,13,9,Times,0,12,0,0,0;6,13,10,Courier,1,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450] Solution :[font = text; inactive; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] The characteristic equation is r^3 - r^2 + r - 1 == 0 which factors as (r-1)(r^2+1) == 0. The roots are thus r==1, r==I, and r==-I. The corresponding fundamental solutions are E^x, Cos[x], and Sin[x]. To satisfy the initial conditions we need: ;[s] 17:0,0;31,1;53,0;71,1;88,0;109,1;114,0;116,1;120,0;126,1;131,0;177,1;180,0;182,1;188,0;194,1;200,0;245,-1; 2:9,13,9,Times,0,12,0,0,0;8,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect] phi[x_] = c1 E^x + c2 Cos[x] + c3 Sin[x]; :[font = input; preserveAspect; startGroup] eqns = {phi[Pi/2]==2, phi'[Pi/2]==1, phi''[Pi/2]==0} :[font = output; output; inactive; preserveAspect; endGroup] {c3 + c1*E^(Pi/2) == 2, -c2 + c1*E^(Pi/2) == 1, -c3 + c1*E^(Pi/2) == 0} ;[o] Pi/2 Pi/2 Pi/2 {c3 + c1 E == 2, -c2 + c1 E == 1, -c3 + c1 E == 0} :[font = input; preserveAspect; startGroup] Solve[eqns] :[font = output; output; inactive; preserveAspect; endGroup] {{c2 -> 0, c3 -> 1, c1 -> E^(-Pi/2)}} ;[o] -Pi/2 {{c2 -> 0, c3 -> 1, c1 -> E }} :[font = text; inactive; preserveAspect] The solution to the initial value problem is :[font = input; preserveAspect; startGroup] phi[x] /. First[%] :[font = output; output; inactive; preserveAspect; endGroup] E^(-Pi/2 + x) + Sin[x] ;[o] -Pi/2 + x E + Sin[x] :[font = input; preserveAspect; startGroup] Plot[%,{x,Pi/2,3Pi/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 -0.452381 0.303152 0.0147151 0.0265847 [ [(1.5)] .00235 .01472 0 2 Msboxa [(2.5)] .3055 .01472 0 2 Msboxa [(3)] .45708 .01472 0 2 Msboxa [(3.5)] .60865 .01472 0 2 Msboxa [(4)] .76023 .01472 0 2 Msboxa [(4.5)] .9118 .01472 0 2 Msboxa [(5)] .14142 .14764 1 0 Msboxa [(10)] .14142 .28056 1 0 Msboxa [(15)] .14142 .41349 1 0 Msboxa [(20)] .14142 .54641 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 .00235 .01472 m .00235 .02097 L s P [(1.5)] .00235 .01472 0 2 Mshowa p .002 w .3055 .01472 m .3055 .02097 L s P [(2.5)] .3055 .01472 0 2 Mshowa p .002 w .45708 .01472 m .45708 .02097 L s P [(3)] .45708 .01472 0 2 Mshowa p .002 w .60865 .01472 m .60865 .02097 L s P [(3.5)] .60865 .01472 0 2 Mshowa p .002 w .76023 .01472 m .76023 .02097 L s P [(4)] .76023 .01472 0 2 Mshowa p .002 w .9118 .01472 m .9118 .02097 L s P [(4.5)] .9118 .01472 0 2 Mshowa p .001 w .03266 .01472 m .03266 .01847 L s P p .001 w .06298 .01472 m .06298 .01847 L s P p .001 w .09329 .01472 m .09329 .01847 L s P p .001 w .12361 .01472 m .12361 .01847 L s P p .001 w .18424 .01472 m .18424 .01847 L s P p .001 w .21455 .01472 m .21455 .01847 L s P p .001 w .24487 .01472 m .24487 .01847 L s P p .001 w .27518 .01472 m .27518 .01847 L s P p .001 w .33581 .01472 m .33581 .01847 L s P p .001 w .36613 .01472 m .36613 .01847 L s P p .001 w .39645 .01472 m .39645 .01847 L s P p .001 w .42676 .01472 m .42676 .01847 L s P p .001 w .48739 .01472 m .48739 .01847 L s P p .001 w .51771 .01472 m .51771 .01847 L s P p .001 w .54802 .01472 m .54802 .01847 L s P p .001 w .57834 .01472 m .57834 .01847 L s P p .001 w .63897 .01472 m .63897 .01847 L s P p .001 w .66928 .01472 m .66928 .01847 L s P p .001 w .6996 .01472 m .6996 .01847 L s P p .001 w .72991 .01472 m .72991 .01847 L s P p .001 w .79054 .01472 m .79054 .01847 L s P p .001 w .82086 .01472 m .82086 .01847 L s P p .001 w .85117 .01472 m .85117 .01847 L s P p .001 w .88149 .01472 m .88149 .01847 L s P p .001 w .94212 .01472 m .94212 .01847 L s P p .001 w .97243 .01472 m .97243 .01847 L s P p .002 w 0 .01472 m 1 .01472 L s P p .002 w .15392 .14764 m .16017 .14764 L s P [(5)] .14142 .14764 1 0 Mshowa p .002 w .15392 .28056 m .16017 .28056 L s P [(10)] .14142 .28056 1 0 Mshowa p .002 w .15392 .41349 m .16017 .41349 L s P [(15)] .14142 .41349 1 0 Mshowa p .002 w .15392 .54641 m .16017 .54641 L s P [(20)] .14142 .54641 1 0 Mshowa p .001 w .15392 .0413 m .15767 .0413 L s P p .001 w .15392 .06788 m .15767 .06788 L s P p .001 w .15392 .09447 m .15767 .09447 L s P p .001 w .15392 .12105 m .15767 .12105 L s P p .001 w .15392 .17422 m .15767 .17422 L s P p .001 w .15392 .20081 m .15767 .20081 L s P p .001 w .15392 .22739 m .15767 .22739 L s P p .001 w .15392 .25398 m .15767 .25398 L s P p .001 w .15392 .30715 m .15767 .30715 L s P p .001 w .15392 .33373 m .15767 .33373 L s P p .001 w .15392 .36032 m .15767 .36032 L s P p .001 w .15392 .3869 m .15767 .3869 L s P p .001 w .15392 .44007 m .15767 .44007 L s P p .001 w .15392 .46666 m .15767 .46666 L s P p .001 w .15392 .49324 m .15767 .49324 L s P p .001 w .15392 .51982 m .15767 .51982 L s P p .001 w .15392 .57299 m .15767 .57299 L s P p .001 w .15392 .59958 m .15767 .59958 L s P p .002 w .15392 0 m .15392 .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 .06788 m .06349 .07138 L .10317 .07493 L .14286 .07865 L .18254 .08262 L .22222 .08696 L .2619 .09182 L .30159 .09736 L .34127 .10376 L .38095 .11124 L .42063 .12002 L .46032 .13038 L .5 .1426 L .53968 .15702 L .57937 .17399 L .61905 .19394 L .65873 .2173 L .69841 .2446 L .7381 .2764 L .77778 .31334 L .81746 .35612 L .85714 .40555 L .89683 .46253 L .93651 .52807 L .97619 .60332 L s 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; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] Find the solution of the initial value problem y''''[x] + 2 y''[x] + y[x] == 3x + 4, y[0] == 0, y'[0] == 0, y''[0] == 1, y'''[0] == 1 by 1) solving the characteristic equation to find fundamental solutions of the homogeneous equation and using the method of undetermined coefficients to find a particular solution; 2) using DSolve[]. Plot the solution. ;[s] 17:0,0;47,1;51,0;52,1;142,0;146,1;150,0;226,1;237,0;311,1;317,0;345,1;349,0;358,1;366,0;367,1;368,0;387,-1; 2:9,13,9,Times,0,12,0,0,0;8,13,10,Courier,1,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450] Solution :[font = text; inactive; preserveAspect] The the characteristic equation is r^4 + 2r^2 + 1 == 0 which factors as (r^2+1)^2 == 0. The roots are thus r1==I, r2==-I, both repeated roots. The corresponding fundamental solutions are Cos[x], Sin[x], x*Cos[x], and x*Sin[x]. Since 0 is not a root of the characteristic equation, the particular solution will have the form A*x + B. Plugging into the equation gives immediately that A = 3 and B = 4. To satisfy the initial conditions we need: ;[s] 26:0,0;35,1;54,0;72,1;86,0;107,1;112,0;114,1;120,0;188,1;194,0;196,1;202,0;204,1;212,0;218,1;226,0;234,1;235,0;325,1;332,0;384,1;389,0;394,1;399,0;444,1;449,-1; 2:13,13,9,Times,0,12,0,0,0;13,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect] phi[x_] = c1 Cos[x] + c2 x*Cos[x] + b1 Sin[x] + b2 x*Sin[x] + 3x + 4; :[font = input; preserveAspect; startGroup] eqns = {phi[0]==0, phi'[0]==0, phi''[0]==1, phi'''[0]==1} :[font = output; output; inactive; preserveAspect; endGroup] {4 + c1 == 0, 3 + b1 + c2 == 0, 2*b2 - c1 == 1, -b1 - 3*c2 == 1} ;[o] {4 + c1 == 0, 3 + b1 + c2 == 0, 2 b2 - c1 == 1, -b1 - 3 c2 == 1} :[font = input; preserveAspect; startGroup] Solve[eqns] :[font = output; output; inactive; preserveAspect; endGroup] {{b2 -> -3/2, c1 -> -4, b1 -> -4, c2 -> 1}} ;[o] 3 {{b2 -> -(-), c1 -> -4, b1 -> -4, c2 -> 1}} 2 :[font = text; inactive; preserveAspect] The solution to the initial value problem is :[font = input; preserveAspect; startGroup] phi[x] /. First[%] :[font = output; output; inactive; preserveAspect; endGroup] 4 + 3*x - 4*Cos[x] + x*Cos[x] - 4*Sin[x] - (3*x*Sin[x])/2 ;[o] 3 x Sin[x] 4 + 3 x - 4 Cos[x] + x Cos[x] - 4 Sin[x] - ---------- 2 :[font = input; preserveAspect; startGroup] Plot[%,{x,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 0.0147151 0.0192405 [ [(2)] .22591 .01472 0 2 Msboxa [(4)] .42801 .01472 0 2 Msboxa [(6)] .63011 .01472 0 2 Msboxa [(8)] .83222 .01472 0 2 Msboxa [(5)] .01131 .11092 1 0 Msboxa [(10)] .01131 .20712 1 0 Msboxa [(15)] .01131 .30332 1 0 Msboxa [(20)] .01131 .39953 1 0 Msboxa [(25)] .01131 .49573 1 0 Msboxa [(30)] .01131 .59193 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 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= subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Problem 7 :[font = text; inactive; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] Given that x, x^2, and 1/x are solutions of the homogeneous equation corresponding to x^3 y'''[x] + x^2 y''[x] - 2x y'[x] + 2y[x] == 2x^4 for x > 0, determine a particular solution. 1) using the method of variation of parameters; 2) using DSolve[]. ;[s] 17:0,0;11,1;12,0;14,1;17,0;23,1;26,0;86,1;140,0;144,1;149,0;184,1;188,0;236,1;240,0;249,1;257,0;259,-1; 2:9,13,9,Times,0,12,0,0,0;8,13,10,Courier,1,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450] Solution :[font = text; inactive; preserveAspect] First, we compute the Wronskian and other determinants needed for Cramer's rule. :[font = input; preserveAspect; startGroup] w = Det[{{ x, x^2, 1/x }, { 1, 2x ,-1/x^2 }, { 0, 2 , 2/x^3 }}] :[font = output; output; inactive; preserveAspect; endGroup] 6/x ;[o] 6 - x :[font = input; preserveAspect; startGroup] w1 = Det[{{ 0, x^2, 1/x }, { 0, 2x ,-1/x^2 }, { 1, 2 , 2/x^3 }}] :[font = output; output; inactive; preserveAspect; endGroup] -3 ;[o] -3 :[font = input; preserveAspect; startGroup] w2 = Det[{{ x, 0 , 1/x }, { 1, 0 ,-1/x^2 }, { 0, 1 , 2/x^3 }}] :[font = output; output; inactive; preserveAspect; endGroup] 2/x ;[o] 2 - x :[font = input; preserveAspect; startGroup] w3 = Det[{{ x, x^2, 0 }, { 1, 2x , 0 }, { 0, 2 , 1 }}] :[font = output; output; inactive; preserveAspect; endGroup] x^2 ;[o] 2 x :[font = text; inactive; preserveAspect] Note that after putting the equation in standard form, the right hand side becomes 2x (not 2x^4 !). A particular solution is then given by the formula (p.208): ;[s] 5:0,0;83,1;85,0;91,1;95,0;160,-1; 2:3,13,9,Times,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] x * Integrate[2x*w1/w,x] + x^2 * Integrate[2x*w2/w,x] + 1/x * Integrate[2x*w3/w,x] :[font = output; output; inactive; preserveAspect; endGroup] x^4/15 ;[o] 4 x -- 15 :[font = text; inactive; preserveAspect] Let's check: :[font = input; preserveAspect; startGroup] x^3 D[x^4/15,{x,3}] + x^2 D[x^4/15,{x,2}] - 2x D[x^4/15,x] + 2 x^4/15 :[font = output; output; inactive; preserveAspect; endGroup] 2*x^4 ;[o] 4 2 x :[font = input; preserveAspect; startGroup] DSolve[x^3y'''[x]+x^2y''[x]-2x*y'[x]+2y[x]==2x^4,y[x],x] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] {{y[x] -> x^4/15 + C[1]/x + x*C[2] + x^2*C[3]}} ;[o] 4 x C[1] 2 {{y[x] -> -- + ---- + x C[2] + x C[3]}} 15 x :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Problem 8 :[font = text; inactive; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] Find approximate values of the solution of the initial value problem y'[t] == 5 t - 3 Sqrt[y[t]], y[0] == 2 at t = 1 1) using the Euler method with h = 0.1; 2) using the Euler method with h = 0.05. Compare these results with the output of NDSolve[] by superimposing the plots of the data. ;[s] 17:0,0;69,1;113,0;116,1;121,0;122,1;126,0;157,1;164,0;166,1;170,0;201,1;209,0;210,1;211,0;252,1;261,0;302,-1; 2:9,13,9,Times,0,12,0,0,0;8,13,10,Courier,1,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450] Solution :[font = text; inactive; preserveAspect] Hints: Define a function that does one step of the Euler method, e.g., EMStep[{t_,y_}] = N[{t+h,y+h*f[t,y]}]; where h is the step size and f[t,y] = 5 t - 3 Sqrt[y]. Then use NestList[EMStep, {t0,y0}, n] to apply EMStep[] successively to its own output, starting at the point {t0,y0} and continuing for n steps. To plot the results use ListPlot[%,PlotJoined->True]. The output of NDSolve[] is a list (in this case a list of one!) of substitution rules of the form {y[t]->InterpolatingFunction[...][t]}. To plot this, do something like sol = NDSolve[...]; Plot[y[t]/.First[sol],{t,0,1}]. You can combine plots with Show[]. ;[s] 27:0,0;71,1;114,0;120,1;121,0;143,1;149,0;150,1;167,0;178,1;206,0;216,1;225,0;279,1;286,0;306,1;307,0;339,1;367,0;384,1;393,0;468,1;505,0;539,1;589,0;618,1;624,0;626,-1; 2:14,13,9,Times,0,12,0,0,0;13,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect] f[t_,y_] = 5 t - 3 Sqrt[y]; :[font = input; preserveAspect] EMStep[{t_,y_}] := N[{t+h,y+h*f[t,y]}] :[font = input; preserveAspect] h = 0.1; :[font = input; preserveAspect; startGroup] soln1=NestList[EMStep,{0,2},10] :[font = output; output; inactive; preserveAspect; endGroup] {{0, 2}, {0.1, 1.575735931288071485}, {0.2, 1.249150969007411062}, {0.3, 1.013854701192412865}, {0.4, 0.8617836448044875461}, {0.5, 0.783286736325409669}, {0.6, 0.7677762727476311185}, {0.7, 0.804907741829061863}, {0.8000000000000000001, 0.8857577896376940719}, {0.9000000000000000001, 1.003413671463802469}, {1., 1.152902056993176355}} ;[o] {{0, 2}, {0.1, 1.57574}, {0.2, 1.24915}, {0.3, 1.01385}, {0.4, 0.861784}, {0.5, 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{0.7500000000000000001, 0.9132492191978441541}, {0.8000000000000000001, 0.9574031085706231117}, {0.8500000000000000002, 1.010632641708794877}, {0.9000000000000000002, 1.072337302130783101}, {0.9500000000000000002, 1.142006721439334517}, {1., 1.219209649624578566}} ;[o] {{0, 2}, {0.05, 1.78787}, {0.1, 1.5998}, {0.15, 1.43508}, {0.2, 1.29288}, {0.25, 1.17233}, {0.3, 1.07242}, {0.35, 0.992079}, {0.4, 0.930175}, {0.45, 0.885506}, {0.5, 0.856854}, {0.55, 0.843005}, {0.6, 0.842782}, {0.65, 0.855077}, {0.7, 0.878871}, {0.75, 0.913249}, {0.8, 0.957403}, {0.85, 1.01063}, {0.9, 1.07234}, {0.95, 1.14201}, {1., 1.21921}} :[font = input; preserveAspect; startGroup] p2 = ListPlot[soln2,PlotJoined->True, 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 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.08966 L .65873 .09354 L .69841 .10397 L .7381 .11778 L .77778 .13476 L .81746 .15474 L .85714 .17756 L .89683 .2031 L Mistroke .93651 .23123 L .97619 .26185 L Mfstroke 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 9 :[font = text; inactive; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] Find approximate values of the solution of the initial value problem y'[t] == 5 t - 3 Sqrt[y[t]], y[0] == 2 at t = 1 1) using the Runge-Kutta method with h = 0.1; 2) using the Runge-Kutta method with h = 0.05. Compare these results with the output of NDSolve[] by superimposing the plots of the data. ;[s] 17:0,0;69,1;113,0;116,1;121,0;122,1;126,0;163,1;170,0;172,1;176,0;213,1;221,0;222,1;223,0;264,1;273,0;314,-1; 2:9,13,9,Times,0,12,0,0,0;8,13,10,Courier,1,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450] Solution :[font = text; inactive; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] Hints: Follow the same ideas as outlined in the hints for Problem 8. Use the program on p.409 of the textbook to create a Runge-Kutta step function: RKStep[{t_,y_}] := ( k1=N[f[t_,y_]]; k2=...; N[{t+h,y+(h/6)*(k1+...+k4)}]) Notice that several calculations can be defined into the function RKStep[] by grouping them together with parentheses and seperating them with semi-colons. The last statement becomes the final output of RKStep[]. ;[s] 7:0,0;149,1;231,0;297,1;305,0;434,1;442,0;444,-1; 2:4,13,9,Times,0,12,0,0,0;3,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect] f[t_,y_] = 5 t - 3 Sqrt[y]; :[font = input; preserveAspect] RKStep[{t_,y_}] := ( k1 = N[f[t,y]]; k2 = N[f[t+0.5*h,y+0.5*h*k1]]; k3 = N[f[t+0.5*h,y+0.5*h*k2]]; k4 = N[f[t+h,y+h*k3]]; N[{t+h,y+(h/6)*(k1+2*k2+2*k3+k4)}] ) :[font = input; preserveAspect] h = 0.1; :[font = input; preserveAspect; startGroup] soln1=NestList[RKStep,{0,2},10] :[font = output; output; inactive; preserveAspect; endGroup] {{0, 2}, {0.1, 1.622307371799809675}, {0.2, 1.333624700148643692}, {0.3, 1.126859401481913322}, {0.4, 0.9938389471251256005}, {0.5, 0.9257254620116500519}, {0.6, 0.9137013543372855866}, {0.7, 0.9497092464571307885}, {0.8000000000000000001, 1.026966781864375039}, {0.9000000000000000001, 1.140127483545222283}, {1., 1.285160551934601151}} ;[o] {{0, 2}, {0.1, 1.62231}, {0.2, 1.33362}, {0.3, 1.12686}, {0.4, 0.993839}, {0.5, 0.925725}, {0.6, 0.913701}, {0.7, 0.949709}, {0.8, 1.02697}, {0.9, 1.14013}, {1., 1.28516}} :[font = input; preserveAspect; startGroup] p1 = ListPlot[soln1,PlotJoined->True, PlotStyle->{RGBColor[0,1,0]}] :[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 0.0238095 0.952381 -0.480368 0.541843 [ [(0.2)] .21429 .06148 0 2 Msboxa [(0.4)] .40476 .06148 0 2 Msboxa [(0.6)] .59524 .06148 0 2 Msboxa [(0.8)] .78571 .06148 0 2 Msboxa [(1)] .97619 .06148 0 2 Msboxa [(1.2)] .01131 .16984 1 0 Msboxa [(1.4)] .01131 .27821 1 0 Msboxa [(1.6)] .01131 .38658 1 0 Msboxa [(1.8)] .01131 .49495 1 0 Msboxa [(2)] .01131 .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 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:[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect] h = 0.05; :[font = input; preserveAspect; startGroup] soln2=NestList[RKStep,{0,2},20] :[font = output; output; inactive; preserveAspect; endGroup] {{0, 2}, {0.05, 1.79962960962333759}, {0.1, 1.622304170690867315}, {0.15, 1.467250250144339559}, {0.2, 1.333618131143278925}, {0.25, 1.220482872663866261}, {0.3, 1.126849549949343595}, {0.35, 1.05166354970494285}, {0.4, 0.9938262737817927566}, {0.45, 0.9522157696920278989}, {0.5, 0.9257108098875404765}, {0.5500000000000000001, 0.9132160821474498589}, {0.6000000000000000001, 0.9136857609100414902}, {0.6500000000000000001, 0.9261429907058674065}, {0.7000000000000000001, 0.9496936505726219165}, {0.7500000000000000001, 0.9835338761878199458}, {0.8000000000000000001, 1.026951828376397175}, {0.8500000000000000002, 1.079324862266541589}, {0.9000000000000000002, 1.140113507767948657}, {0.9500000000000000002, 1.208853602744947435}, {1., 1.285147666473209233}} ;[o] {{0, 2}, {0.05, 1.79963}, {0.1, 1.6223}, {0.15, 1.46725}, {0.2, 1.33362}, {0.25, 1.22048}, {0.3, 1.12685}, {0.35, 1.05166}, {0.4, 0.993826}, {0.45, 0.952216}, {0.5, 0.925711}, {0.55, 0.913216}, {0.6, 0.913686}, {0.65, 0.926143}, {0.7, 0.949694}, {0.75, 0.983534}, {0.8, 1.02695}, {0.85, 1.07932}, {0.9, 1.14011}, {0.95, 1.20885}, {1., 1.28515}} :[font = input; preserveAspect; startGroup] p2 = ListPlot[soln2,PlotJoined->True, 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 0.0238095 0.952381 -0.479884 0.541601 [ [(0.2)] .21429 .06172 0 2 Msboxa [(0.4)] .40476 .06172 0 2 Msboxa [(0.6)] .59524 .06172 0 2 Msboxa [(0.8)] .78571 .06172 0 2 Msboxa [(1)] .97619 .06172 0 2 Msboxa 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t*Sin[3t] by integrating, Integrate[E^(-s*t)*f[t],{t,0,Infinity}], and by using LaplaceTransform[f[t],t,s]. Find the inverse Laplace transform of the functions c) 3s/(s^2 - s - 6) d) (2s-3)/(s^2 + 2s + 10) by using partial fraction decompositions (use Apart[]) if necessary and the tables on p.286 of the textbook, and by using InverseLaplaceTransform[Y[s],s,t]. ;[s] 23:0,0;45,1;49,2;51,1;61,0;62,1;66,2;68,1;80,0;96,1;135,0;150,1;176,0;230,1;234,2;236,1;259,2;261,1;286,0;332,1;339,0;408,1;441,0;443,-1; 3:8,13,9,Times,0,12,0,0,0;11,13,10,Courier,1,12,0,0,0;4,13,9,Times,2,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450] Solution :[font = text; inactive; preserveAspect] a) ;[s] 1:0,1;3,-1; 2:0,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0; :[font = input; preserveAspect] f1[t_] = t*E^(2t); :[font = input; preserveAspect; startGroup] Integrate[E^(-s*t)*f1[t],{t,0,Infinity}] :[font = output; output; inactive; preserveAspect; endGroup] (-2 + s)^(-2) ;[o] -2 (-2 + s) :[font = input; preserveAspect; startGroup] Y1[s_] = LaplaceTransform[f1[t],t,s] :[font = output; output; inactive; preserveAspect; endGroup] (-2 + s)^(-2) ;[o] -2 (-2 + s) :[font = text; inactive; preserveAspect] b) ;[s] 1:0,0;3,-1; 1:1,13,9,Times,2,12,0,0,0; :[font = input; preserveAspect] f2[t_] = t*Sin[3t]; :[font = input; preserveAspect; startGroup] Integrate[E^(-s*t)*f2[t],{t,0,Infinity}] :[font = output; output; inactive; preserveAspect; endGroup] (6*s)/(9 + s^2)^2 ;[o] 6 s --------- 2 2 (9 + s ) :[font = input; preserveAspect; startGroup] LaplaceTransform[f2[t],t,s] :[font = output; output; inactive; preserveAspect; endGroup] (6*s)/(9 + s^2)^2 ;[o] 6 s --------- 2 2 (9 + s ) :[font = text; inactive; preserveAspect] c) ;[s] 1:0,0;3,-1; 1:1,13,9,Times,2,12,0,0,0; :[font = input; preserveAspect; startGroup] Apart[3s/(s^2 - s - 6)] :[font = output; output; inactive; preserveAspect; endGroup] 9/(5*(-3 + s)) + 6/(5*(2 + s)) ;[o] 9 6 ---------- + --------- 5 (-3 + s) 5 (2 + s) :[font = text; inactive; preserveAspect] The first term contributes (from p.286) 9/5 E^(3t) and the second terms contributes 6/5 E^(-2t) so f3[t_] = 9/5 E^(3t) + 6/5 E^(-2t). ;[s] 9:0,0;40,1;50,0;84,1;95,0;99,1;122,0;125,1;136,0;138,-1; 2:5,13,9,Times,0,12,0,0,0;4,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] InverseLaplaceTransform[3s/(s^2 - s - 6),s,t] :[font = output; output; inactive; preserveAspect; endGroup] 6/(5*E^(2*t)) + (9*E^(3*t))/5 ;[o] 3 t 6 9 E ------ + ------ 2 t 5 5 E :[font = text; inactive; preserveAspect] d) First complete the square in the demnominator and rewrite the numerator: (2s-3)/(s^2+2s+10) == 2(s+1)/((s+1)^2+3^2) - 5/((s+1)^2+3^2) The term 2(s+1)/((s+1)^2+3^2) contributes (from p.286) 2E^(-t)*Cos[3t], and the second term -5/((s+1)^2 + 3^2) contributes -5/3 E^(-t)Sin[3t], so f4[t_] = 2E^(-t)*Cos[3t] - 5/3 E^(-t)Sin[3t] ;[s] 13:0,0;2,1;77,2;139,1;148,2;168,1;194,2;209,1;231,2;249,1;262,2;280,1;285,2;334,-1; 3:1,13,9,Times,2,12,0,0,0;6,13,9,Times,0,12,0,0,0;6,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] InverseLaplaceTransform[(2s-3)/(s^2 + 2s + 10),s,t] :[font = output; output; inactive; preserveAspect; endGroup] -(Sin[3*t]/E^t) + 2*(Cos[3*t]/E^t - Sin[3*t]/(3*E^t)) ;[o] Sin[3 t] Cos[3 t] Sin[3 t] -(--------) + 2 (-------- - --------) t t t E E 3 E :[font = input; preserveAspect; startGroup] Simplify[%] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] (6*Cos[3*t] - 5*Sin[3*t])/(3*E^t) ;[o] 6 Cos[3 t] - 5 Sin[3 t] ----------------------- t 3 E ^*)