(*^ ::[ 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 = "NeXT Mathematica Notebook Front End Version 2.2"; NeXTStandardFontEncoding; fontset = title, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e8, 24, "Times"; ; fontset = subtitle, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e6, 18, "Times"; ; fontset = subsubtitle, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, L1, e6, 14, "Times"; ; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, L1, a20, 18, "Times"; ; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, L1, a15, 14, "Times"; ; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, L1, a12, 12, "Times"; ; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 10, "Times"; ; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L1, 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, L1, 12, "Courier"; ; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, L1, 12, "Courier"; ; fontset = name, inactive, noPageBreakInGroup, nohscroll, preserveAspect, M7, italic, B65535, L1, 10, "Times"; ; fontset = header, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, L1, 12, "Times"; ; fontset = leftheader, 12; fontset = footer, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M7, italic, L1, 12, "Times"; ; fontset = leftfooter, 12; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12, "Courier"; ; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; paletteColors = 128; showRuler; automaticGrouping; currentKernel; ] :[font = text; inactive; preserveAspect; center] Linear, Constant Coefficient, Homogeneous Differential Equations ;[s] 1:0,0;65,-1; 1:1,21,16,Times,1,24,0,0,0; :[font = subtitle; inactive; preserveAspect; startGroup] Steve Dunbar Department of Mathematics and Statistics University of Nebraska-Lincoln Lincoln, NE 68588-0323 sdunbar@mathlab01.unl.edu Fall Semester 1993 :[font = subsubtitle; inactive; preserveAspect] Covers Nagel & Saff, Sections 4.6, 6.2, 6.3 :[font = section; inactive; preserveAspect; startGroup] Before Starting :[font = text; inactive; preserveAspect] Define a new Mathematica function to compute the Wronskian of two functions. ;[s] 3:0,0;13,1;24,2;77,-1; 3:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect; endGroup] Wronskian[f1_, f2_] := Det[{{f1,f2},{D[f1,x], D[f2,x]}}] :[font = section; inactive; preserveAspect; startGroup] Fundamental Sets of Solutions :[font = text; inactive; preserveAspect] Use the Mathematica command DSolve to analytically solve second order differential equations. The solution returns as a solution rule. From the solution rule, we can read off the fundmental set of solutions. ;[s] 3:0,0;8,1;19,2;212,-1; 3:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect] DSolve[ 2*y''[x] + 9*y'[x] + 9*y[x] == 0, y[x],x] :[font = text; inactive; preserveAspect] From this, we can see that the fundamental set of solutions is Exp[-3 x] and Exp[-3 x/2] :[font = input; preserveAspect; startGroup] Wronskian[ Exp[-3*x], Exp[-3*x/2] ] :[font = output; output; inactive; preserveAspect; endGroup] 3/(2*E^((9*x)/2)) ;[o] 3 ---------- (9 x)/2 2 E :[font = text; inactive; preserveAspect] We can also look at the auxiliary equation, and solve directly for the eignevalues: :[font = input; preserveAspect; startGroup] Solve[ 2*r^2 + 9*r + 9 == 0, r] :[font = output; output; inactive; preserveAspect; endGroup] {{r -> -3}, {r -> -3/2}} ;[o] 3 {{r -> -3}, {r -> -(-)}} 2 :[font = input; preserveAspect] Factor[2*r^2 + 9*r + 9 ] :[font = subsection; inactive; preserveAspect; startGroup] Experiments to Perform :[font = text; inactive; preserveAspect; plain; bold; fontName = "Times"] (Borelli and Coleman, Experiment 4.2.3 , page 122) Consider the homogeneous, constant coeffiicient equation y'' - y' + b*y == 0. Solve y(x) for values of b = -1, 0, 1/4, 1 . Find the fundametnal set of solutions in each case. Evaluate the Wronskian in each case. Find the eigenvalues by solving the auxiliary equation in each case. :[font = text; inactive; preserveAspect; plain; bold; fontName = "Times"; endGroup; endGroup] (Borelli and Coleman, Experiment 4.2.4, page 122) Consider the homogeneous, constant coefficient equation y'' - a*y' - y == 0 for a = -1, 0, 1/4, 1. Find the fundametnal set of solutions in each case. Evaluate the Wronskian in each case. Find the eigenvalues by solving the auxiliary equation in each case. :[font = section; inactive; preserveAspect; startGroup] Higher Order Equations :[font = text; inactive; preserveAspect; plain; bold; fontName = "Times"] (Nagle & Saff, Section6.3, page 258, #8) Solve the third-order, constant coefficient, homogeneous differential equation y''' +5 y'' -13 y' + 7 y = 0 with DSolve. Find the eigenvalues by using the Solve command on the auxiliary equation. Find the eigenvalues by using the Factor command on the auxiliary polynomial. ;[s] 3:0,0;134,1;135,2;321,-1; 3:1,10,8,Times,1,12,0,0,0;1,0,0,Symbol,0,12,0,0,0;1,10,8,Times,1,12,0,0,0; :[font = text; inactive; preserveAspect; plain; bold; fontName = "Times"; endGroup; endGroup] (Nagle & Saff, Section6.3, page 259, #25) Solve the third-order, constant coefficient, homogeneous differential equation y''' +5 y'' -13 y' + 7 y = 0 with DSolve. Find the eigenvalues by using the Solve command on the auxiliary equation. Find the eigenvalues by using the Factor command on the auxiliary polynomial. What happens? Try to estimate the eigenvalues by graphing the auxiliary polynomial and observing the root crossings. ;[s] 3:0,0;135,1;136,2;438,-1; 3:1,10,8,Times,1,12,0,0,0;1,0,0,Symbol,0,12,0,0,0;1,10,8,Times,1,12,0,0,0; ^*)