(*^ ::[ 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; automaticGrouping; currentKernel; ] :[font = subsubsection; inactive; preserveAspect; startGroup] Assignment #24, Problem 15 (p.456) :[font = input; initialization; noKeepOnOnePage; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] *) < 0, y -> 0}, {y -> 0, x -> -I/2^(1/2)}, {y -> 0, x -> I/2^(1/2)}} ;[o] -I {{x -> 0, y -> 0}, {y -> 0, x -> -------}, Sqrt[2] I {y -> 0, x -> -------}} Sqrt[2] :[font = input; preserveAspect; startGroup] A = {{D[F[x,y],x], D[F[x,y],y]}, {D[G[x,y],x], D[G[x,y],y]}}; A//MatrixForm :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{0, 1}, {1 + 6*x^2, 0}}] ;[o] 0 1 2 1 + 6 x 0 :[font = input; preserveAspect; startGroup] A1 = A /. cp[[1]]; A1//MatrixForm :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{0, 1}, {1, 0}}] ;[o] 0 1 1 0 :[font = input; preserveAspect; startGroup] Eigensystem[A1] :[font = output; output; inactive; preserveAspect; endGroup] {{-1, 1}, {{-1, 1}, {1, 1}}} ;[o] {{-1, 1}, {{-1, 1}, {1, 1}}} :[font = text; inactive; preserveAspect] Eigenvalues of opposite sign imply a saddle point. :[font = text; inactive; preserveAspect] b) Sketch the trajectories for the corresponding linear system by integrating the equation for y'[x]. Show from the parametric form of the solution that the only trajectory on which x->0, y->0 as t->Infinity is y == -x. ;[s] 12:0,0;2,1;96,2;101,1;183,2;187,1;189,2;193,1;197,2;208,1;212,2;219,1;221,-1; 3:1,13,9,Times,2,12,0,0,0;6,13,9,Times,0,12,0,0,0;5,13,10,Courier,1,12,0,0,0; :[font = text; inactive; preserveAspect] The corresponding linear system is x'[t] == y[t] y'[t] == x[t] So the trajectories are defined by the the equation y'[x] == x/y[x] which can easily be integrated to get the equation of a family of hyperbolas. ;[s] 5:0,0;35,1;71,0;123,1;138,0;217,-1; 2:3,13,9,Times,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] DSolve[ y'[x] == x/y[x], y[x], x] :[font = output; output; inactive; preserveAspect; endGroup] {{y[x] -> -(x^2 + C[1])^(1/2)}, {y[x] -> (x^2 + C[1])^(1/2)}} ;[o] 2 2 {{y[x] -> -Sqrt[x + C[1]]}, {y[x] -> Sqrt[x + C[1]]}} :[font = input; preserveAspect] trajs = Flatten[Table[{-(x^2 + c)^(1/2), (x^2 + c)^(1/2)}, {c,-5,5}]]; :[font = 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preserveAspect; endGroup] {c1/E^t + c2*E^t, -(c1/E^t) + c2*E^t} ;[o] c1 t c1 t {-- + c2 E , -(--) + c2 E } t t E E :[font = text; inactive; preserveAspect] The only trajectories that approaches {0,0} are the ones for which c2 = 0, i.e., for ;[s] 5:0,0;38,1;43,0;67,1;73,0;85,-1; 2:3,13,9,Times,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect] X1[t_] = c1 {1,-1} E^(-t) :[font = text; inactive; preserveAspect] On these trajectories the y coordinate always equals -x, and so the trajectories lie on the line y == -x. It is clear that both coordinates go to 0 as t->Infinity. ;[s] 11:0,0;26,1;27,0;53,1;55,0;97,1;104,0;146,1;147,0;151,1;162,0;164,-1; 2:6,13,9,Times,0,12,0,0,0;5,13,10,Courier,1,12,0,0,0; :[font = text; inactive; preserveAspect] c) Determine the trajectories for the nonlinear system by integrating the equation for y'[x]. Sketch the trajectories for the nonlinear system that correspond to y==x and y==-x for the linear system. ;[s] 8:0,0;2,1;88,2;93,1;163,2;167,1;172,2;177,1;201,-1; 3:1,13,9,Times,2,12,0,0,0;4,13,9,Times,0,12,0,0,0;3,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] y'[x] == G[x,y[x]]/F[x,y[x]] :[font = output; output; inactive; preserveAspect; endGroup] Derivative[1][y][x] == (x + 2*x^3)/y[x] ;[o] 3 x + 2 x y'[x] == -------- y[x] :[font = input; preserveAspect; startGroup] DSolve[ y'[x] == G[x,y[x]]/F[x,y[x]], y[x], x ] :[font = output; output; inactive; preserveAspect; endGroup] {{y[x] -> -(x^2 + x^4 + C[1])^(1/2)}, {y[x] -> (x^2 + x^4 + C[1])^(1/2)}} ;[o] 2 4 {{y[x] -> -Sqrt[x + x + C[1]]}, 2 4 {y[x] -> Sqrt[x + x + C[1]]}} :[font = input; preserveAspect] trajs = Flatten[Table[{-(x^2 + x^4 + c)^(1/2), (x^2 + x^4 + c)^(1/2)}, {c,-5,5}]]; :[font = input; preserveAspect; startGroup] Plot[Evaluate[trajs], {x,-3,3}, PlotRange->{-5,5}] :[font = postscript; PostScript; 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