(*^ ::[ 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, a12, 12, "Times"; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, "Times"; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L-5, 12, "Courier"; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, R65535, L-5, 12, "Courier"; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, B65535, L-5, 12, "Courier"; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, 12, "Courier"; fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, 10, "Geneva"; fontset = header, inactive, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = leftheader, inactive, L2, 12, "Times"; fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M7, 12, "Times"; fontset = leftfooter, inactive, L2, 12, "Times"; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, "Times"; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; paletteColors = 128; showRuler; currentKernel; ] :[font = subsection; inactive; preserveAspect; rightWrapOffset = 624] Math 325: Differential Equations Quiz 4 11/8/96 :[font = subsubsection; inactive; preserveAspect; rightWrapOffset = 624] 1. The solution to the initial value problem, X'[t] == A.X[t], X[0] == X0, is given by X[t] = Exp[A*t].X0 where Exp[A*t] = I + A*t + A^2*(t^2/2!) + A^3*(t^3/3!) + ... Let A = {{3,-4}, {5,-6}}; Calculate the matrix Exp[A*t] another way using the eigenvalues and eigenvectors of A. ;[s] 13:0,0;46,1;74,0;87,1;105,0;112,1;167,0;171,1;205,0;227,1;235,0;290,1;291,0;293,-1; 2:7,13,9,Times,1,12,0,0,0;6,13,10,Courier,1,12,0,0,0; :[font = subsubsection; inactive; preserveAspect; rightWrapOffset = 624; startGroup] Solution :[font = input; preserveAspect; rightWrapOffset = 624] A = {{3,-4}, {5,-6}}; ;[s] 2:0,0;26,1;28,-1; 2:1,12,10,Courier,1,12,0,0,0;1,12,9,Times,1,12,0,0,0; :[font = text; inactive; preserveAspect; rightWrapOffset = 624] The matrix Exp[A*t] is the unique fundamental matrix Phi[t] that satisfies Phi[0] == Identity[2]. This matrix Phi[t] can be calculated as Psi[t].Inverse[Psi[0]] where Psi[t] is any fundamental matrix, i.e., a matrix whose columns form a fundamental solution of the above equation. To get such a fundamental solution we need the eigenvalues and eigenvectors of A. ;[s] 16:0,0;11,1;19,0;53,1;59,0;75,1;96,2;98,0;110,1;116,0;138,1;160,0;167,1;173,0;360,1;361,0;363,-1; 3:8,13,9,Times,0,12,0,0,0;7,13,10,Courier,1,12,0,0,0;1,13,9,Times,1,12,0,0,0; :[font = input; preserveAspect; rightWrapOffset = 624; startGroup] Eigensystem[A] :[font = output; output; inactive; preserveAspect; rightWrapOffset = 624; endGroup] {{-2, -1}, {{4, 5}, {1, 1}}} ;[o] {{-2, -1}, {{4, 5}, {1, 1}}} :[font = text; inactive; preserveAspect; rightWrapOffset = 624] Two independent solutions are thus :[font = input; preserveAspect; rightWrapOffset = 624] X1[t_] = {4,5}Exp[-2t]; X2[t_] = {1,1}Exp[-t]; :[font = text; inactive; preserveAspect; rightWrapOffset = 624] A fundamental matrix is given by (the transpose turns X1[t] and X2[t] into columns): ;[s] 5:0,0;54,1;59,0;64,1;69,0;85,-1; 2:3,13,9,Times,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; rightWrapOffset = 624; startGroup] Psi[t_] = Transpose[{X1[t], X2[t]}]; Psi[t]//MatrixForm :[font = output; output; inactive; preserveAspect; rightWrapOffset = 624; endGroup] MatrixForm[{{4/E^(2*t), E^(-t)}, {5/E^(2*t), E^(-t)}}] ;[o] 4 ---- 2 t -t E E 5 ---- 2 t -t E E :[font = input; preserveAspect; rightWrapOffset = 624; startGroup] Psi[0]//MatrixForm :[font = output; output; inactive; preserveAspect; rightWrapOffset = 624; endGroup] MatrixForm[{{4, 1}, {5, 1}}] ;[o] 4 1 5 1 :[font = input; preserveAspect; rightWrapOffset = 624; startGroup] Inverse[Psi[0]]//MatrixForm :[font = output; output; inactive; preserveAspect; rightWrapOffset = 624; endGroup] MatrixForm[{{-1, 1}, {5, -4}}] ;[o] -1 1 5 -4 :[font = text; inactive; preserveAspect; rightWrapOffset = 624] The matrix Exp[A*t] is then given by ;[s] 3:0,0;11,1;19,0;37,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; rightWrapOffset = 624; startGroup] Phi[t_] = Psi[t].Inverse[Psi[0]]; Phi[t]//MatrixForm :[font = output; output; inactive; preserveAspect; rightWrapOffset = 624; endGroup] MatrixForm[{{-4/E^(2*t) + 5/E^t, 4/E^(2*t) - 4/E^t}, {-5/E^(2*t) + 5/E^t, 5/E^(2*t) - 4/E^t}}] ;[o] -4 5 4 4 ---- + -- ---- - -- 2 t t 2 t t E E E E -5 5 5 4 ---- + -- ---- - -- 2 t t 2 t t E E E E :[font = text; inactive; preserveAspect; rightWrapOffset = 624] To see that this really is the result of summing the series, let's add a few terms: :[font = input; preserveAspect; rightWrapOffset = 624] expA[n_]=Sum[MatrixPower[A,i]*(t^i/i!),{i,0,n}]; :[font = input; preserveAspect; rightWrapOffset = 624; startGroup] expA[1]//MatrixForm :[font = output; output; inactive; preserveAspect; rightWrapOffset = 624; endGroup] MatrixForm[{{1 + 3*t, -4*t}, {5*t, 1 - 6*t}}] ;[o] 1 + 3 t -4 t 5 t 1 - 6 t :[font = input; preserveAspect; rightWrapOffset = 624; startGroup] expA[2]//MatrixForm :[font = output; output; inactive; preserveAspect; rightWrapOffset = 624; endGroup] MatrixForm[{{1 + 3*t - (11*t^2)/2, -4*t + 6*t^2}, {5*t - (15*t^2)/2, 1 - 6*t + 8*t^2}}] ;[o] 2 11 t 1 + 3 t - ----- 2 2 -4 t + 6 t 2 15 t 5 t - ----- 2 2 1 - 6 t + 8 t :[font = input; preserveAspect; rightWrapOffset = 624; startGroup] expA[3]//MatrixForm :[font = output; output; inactive; preserveAspect; rightWrapOffset = 624; endGroup] MatrixForm[{{1 + 3*t - (11*t^2)/2 + (9*t^3)/2, -4*t + 6*t^2 - (14*t^3)/3}, {5*t - (15*t^2)/2 + (35*t^3)/6, 1 - 6*t + 8*t^2 - 6*t^3}}] ;[o] 2 3 3 11 t 9 t 2 14 t 1 + 3 t - ----- + ---- -4 t + 6 t - ----- 2 2 3 2 3 15 t 35 t 5 t - ----- + ----- 2 3 2 6 1 - 6 t + 8 t - 6 t :[font = text; inactive; preserveAspect; rightWrapOffset = 624] Compare with three terms of the series expansion about the point 0. ;[s] 3:0,0;65,1;66,0;68,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; rightWrapOffset = 624; startGroup] Series[Phi[t],{t,0,3}]//MatrixForm :[font = output; output; inactive; preserveAspect; rightWrapOffset = 624; endGroup] MatrixForm[{{SeriesData[t, 0, {1, 3, -11/2, 9/2}, 0, 4, 1], SeriesData[t, 0, {-4, 6, -14/3}, 1, 4, 1]}, {SeriesData[t, 0, {5, -15/2, 35/6}, 1, 4, 1], SeriesData[t, 0, {1, -6, 8, -6}, 0, 4, 1]}}] ;[o] 2 3 3 11 t 9 t 4 2 14 t 4 1 + 3 t - ----- + ---- + O[t] -4 t + 6 t - ----- + O[t] 2 2 3 2 3 15 t 35 t 4 5 t - ----- + ----- + O[t] 2 3 4 2 6 1 - 6 t + 8 t - 6 t + O[t] :[font = text; inactive; preserveAspect; rightWrapOffset = 624] Mathematica can perform the matrix exponentiation in one step: :[font = input; preserveAspect; rightWrapOffset = 624; startGroup] MatrixExp[A*t]//MatrixForm :[font = output; output; inactive; preserveAspect; rightWrapOffset = 624; endGroup; endGroup] MatrixForm[{{-4/E^(2*t) + 5/E^t, 4/E^(2*t) - 4/E^t}, {-5/E^(2*t) + 5/E^t, 5/E^(2*t) - 4/E^t}}] ;[o] -4 5 4 4 ---- + -- ---- - -- 2 t t 2 t t E E E E -5 5 5 4 ---- + -- ---- - -- 2 t t 2 t t E E E E :[font = subsubsection; inactive; preserveAspect; rightWrapOffset = 624; startGroup] 2. A fundamental solution for the homogeneous linear system X'[t] == P[t].X[t] is X[t] = c1 {2,-1} t^2 + c2 {-1,1} t^3 Use the method of variation of parameters to find a particular solution to the non-homogeneous equation X'[t] == P[t].X[t] + {t,-t} ;[s] 6:0,0;61,1;84,0;86,1;129,0;233,1;265,-1; 2:3,19,14,Times,1,12,0,0,0;3,18,13,Courier,1,12,0,0,0; :[font = subsubsection; inactive; preserveAspect; rightWrapOffset = 624; endGroup] Solution :[font = text; inactive; preserveAspect; rightWrapOffset = 624] A fundamental matrix is built from the given fundamental solutions: :[font = input; preserveAspect; rightWrapOffset = 624; startGroup] Psi[t_] = Transpose[{{2t^2,-t^2},{-t^3,t^3}}]; Psi[t]//MatrixForm :[font = output; output; inactive; preserveAspect; rightWrapOffset = 624; endGroup] MatrixForm[{{2*t^2, -t^3}, {-t^2, t^3}}] ;[o] 2 3 2 t -t 2 3 -t t :[font = input; preserveAspect; rightWrapOffset = 624; startGroup] Inverse[Psi[t]]//MatrixForm :[font = output; output; inactive; preserveAspect; rightWrapOffset = 624; endGroup] MatrixForm[{{t^(-2), t^(-2)}, {t^(-3), 2/t^3}}] ;[o] -2 -2 t t 2 -- -3 3 t t :[font = text; inactive; preserveAspect; rightWrapOffset = 624] A particular solution is given by Y[t] = Psi[t].U[t] where U[t] satisfies: ;[s] 5:0,0;34,1;52,0;59,1;63,0;75,-1; 2:3,13,9,Times,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; rightWrapOffset = 624; startGroup] U'[t] == Inverse[Psi[t]].{t,-t} :[font = output; output; inactive; preserveAspect; rightWrapOffset = 624; endGroup] Derivative[1][U][t] == {0, -t^(-2)} ;[o] -2 U'[t] == {0, -t } :[font = input; preserveAspect; rightWrapOffset = 624; startGroup] U[t_] = Integrate[Inverse[Psi[t]].{t,-t},t] :[font = output; output; inactive; preserveAspect; rightWrapOffset = 624; endGroup] {0, t^(-1)} ;[o] 1 {0, -} t :[font = text; inactive; preserveAspect; rightWrapOffset = 624] A particular solution is then :[font = input; preserveAspect; rightWrapOffset = 624; startGroup] Y[t_] = Psi[t].U[t] :[font = output; output; inactive; preserveAspect; rightWrapOffset = 624; endGroup] {-t^2, t^2} ;[o] 2 2 {-t , t } ^*)