(*^ ::[ 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-4, 12, "Courier"; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-4, 12, "Courier"; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, R32768, L-4, 12, "Courier"; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-4, 12, "Courier"; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, B32768, L-4, 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; currentKernel; ] :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] Math 325: Differential Equations Assignment 2 :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] Name: :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] Section: :[font = text; inactive; preserveAspect; startGroup] I affirm that the solutions presented in this assignment are entirely my own work. :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450; endGroup] Signature: :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Initialization :[font = input; initialization; noKeepOnOnePage; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450; endGroup] *) <=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] Solution :[font = input; preserveAspect; endGroup] :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Problem 3 :[font = text; inactive; preserveAspect] 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] Solution :[font = input; preserveAspect; endGroup] :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Problem 4 :[font = text; inactive; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] 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] Solution :[font = input; preserveAspect; endGroup] :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Problem 5 :[font = text; inactive; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] 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] Solution :[font = input; preserveAspect; endGroup] :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Problem 6 :[font = text; inactive; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] 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;257,1;262,0;270,2;275,0;281,1;290,0;298,2;311,0;329,1;345,0;383,1;390,0;438,2;440,0;473,1;568,0;622,2;624,0;651,1;655,0;657,-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] Solution :[font = input; preserveAspect; endGroup] :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Problem 7 :[font = text; inactive; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] a) Compute the inverse of the 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] Solution :[font = input; preserveAspect; endGroup] :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Problem 8 :[font = text; inactive; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] 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} == {0,0,0} 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;274,0;312,1;325,0;327,2;329,0;344,1;345,0;356,1;375,0;382,1;383,0;441,1;442,0;458,1;472,0;474,-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 = input; preserveAspect; endGroup] :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Problem 9 :[font = text; inactive; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] 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] Solution :[font = input; preserveAspect; endGroup] :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Problem 10 :[font = text; inactive; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] Find the general solution of the system x1'[t] == x1[t] - 2 x2[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] Solution :[font = input; preserveAspect; endGroup] ^*)