(*^ ::[ 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] Math 325: Differential Equations Quiz 3 10/16/96 :[font = subsubsection; inactive; preserveAspect] 1. Transform the initial value problem y'''[t] - t y'[t] + t^3 y[t] == Cos[t], y[0]==1, y'[0]==0, y''[0]==-2 into a system of first order equations with initial conditions. ;[s] 3:0,0;39,1;117,0;181,-1; 2:2,13,9,Times,1,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = subsubsection; inactive; preserveAspect; startGroup] Solution :[font = text; inactive; preserveAspect] Let :[font = input; preserveAspect] x1[t] == y[t] x2[t] == x1'[t] == y'[t] x3[t] == x2'[t] == y''[t] x3'[t] == y'''[t] == t y'[t] - t^3 y[t] + Cos[t] ;[s] 5:0,0;13,1;14,0;38,1;39,0;114,-1; 2:3,12,10,Courier,1,12,0,0,0;2,12,9,Times,0,12,0,0,0; :[font = text; inactive; preserveAspect] The corresponding system is then :[font = input; preserveAspect] x1'[t] == x2[t] x2'[t] == x3[t] x3'[t] == t x2[t] - t^3 x1[t] + Cos[t] :[font = text; inactive; preserveAspect] The initial conditions y[0]==1, y'[0]==0, y''[0]==-2 become ;[s] 3:0,0;23,1;52,0;60,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; endGroup] x1[0] == 1, x2[0] == 0, x3[0]==-2 :[font = subsubsection; inactive; preserveAspect; startGroup] 2. Find the eigenvalues and corresponding eigenvectors of the matrix {{-11, 5}, {-30, 14}} ;[s] 2:0,0;70,1;102,-1; 2:1,13,9,Times,1,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = subsubsection; inactive; preserveAspect; endGroup] Solution :[font = input; preserveAspect; startGroup] A = {{-11, 5}, {-30, 14}}; :[font = text; inactive; preserveAspect] The eigenvalues are the roots of the polynomial given by :[font = input; preserveAspect; startGroup] Det[A - r IdentityMatrix[2]] :[font = output; output; inactive; preserveAspect; endGroup] -4 - 3*r + r^2 ;[o] 2 -4 - 3 r + r :[font = input; preserveAspect; startGroup] Solve[% == 0] :[font = output; output; inactive; preserveAspect; endGroup] {{r -> -1}, {r -> 4}} ;[o] {{r -> -1}, {r -> 4}} :[font = text; inactive; preserveAspect] To find an eigenvector corresponding to the eigenvalue -1, we must solve ;[s] 3:0,0;55,1;57,0;73,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] (A - (-1)IdentityMatrix[2]).{v1,v2} == {0,0} :[font = output; output; inactive; preserveAspect; endGroup] {-10*v1 + 5*v2, -30*v1 + 15*v2} == {0, 0} ;[o] {-10 v1 + 5 v2, -30 v1 + 15 v2} == {0, 0} :[font = text; inactive; preserveAspect] We solve this by hand using row reduction. :[font = input; preserveAspect; startGroup] RowReduce[A - (-1)IdentityMatrix[2]] //MatrixForm :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{1, -1/2}, {0, 0}}] ;[o] 1 -(-) 1 2 0 0 :[font = text; inactive; preserveAspect] In one step: :[font = input; preserveAspect; startGroup] Solve[(A - (-1)IdentityMatrix[2]).{v1,v2} == {0,0}] :[font = output; output; inactive; preserveAspect; endGroup] {{v1 -> v2/2}} ;[o] v2 {{v1 -> --}} 2 :[font = text; inactive; preserveAspect] So, one eigenvector for the eigenvlaue -1 is V == {1, 2} ;[s] 4:0,0;39,1;41,0;45,1;57,-1; 2:2,13,9,Times,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0; :[font = text; inactive; preserveAspect] To find an eigenvector corresponding to the other eigenvalue 4, we solve ;[s] 3:0,0;61,1;62,0;73,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] (A - 4 IdentityMatrix[2]).{v1,v2} == {0,0} :[font = output; output; inactive; preserveAspect; endGroup] {-15*v1 + 5*v2, -30*v1 + 10*v2} == 0 ;[o] {-15 v1 + 5 v2, -30 v1 + 10 v2} == 0 :[font = input; preserveAspect; startGroup] RowReduce[A - 4 IdentityMatrix[2]] //MatrixForm :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{1, -1/3}, {0, 0}}] ;[o] 1 -(-) 1 3 0 0 :[font = input; preserveAspect; startGroup] Solve[(A - 4 IdentityMatrix[2]).{v1,v2} == {0,0}] :[font = output; output; inactive; preserveAspect; endGroup] {{v1 -> v2/3}} ;[o] v2 {{v1 -> --}} 3 :[font = text; inactive; preserveAspect] So, an eigenvector for the eigenvalue 4 is V == {1, 3}. Let's check these answers: ;[s] 5:0,0;38,1;39,0;43,1;54,0;83,-1; 2:3,13,9,Times,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] A.{1,2} :[font = output; output; inactive; preserveAspect; endGroup] {-1, -2} ;[o] {-1, -2} :[font = input; preserveAspect; startGroup] A.{1,3} :[font = output; output; inactive; preserveAspect; endGroup; endGroup] {4, 12} ;[o] {4, 12} ^*)