(*^ ::[ 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 1 9/11/96 :[font = input; preserveAspect] L[y_] := D[y,{x,4}] + 4D[y,{x,3}] + 13 D[y,{x,2}] :[font = subsubsection; inactive; preserveAspect] a) Find the general solution of the differential equation L[y] == 0. ;[s] 4:0,1;2,0;58,2;67,0;69,-1; 3:2,13,9,Times,1,12,0,0,0;1,13,9,Times,3,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = subsubsection; inactive; preserveAspect] Solution :[font = input; preserveAspect; startGroup] Solve[r^4 + 4r^3 + 13r^2==0] :[font = output; output; inactive; preserveAspect; endGroup] {{r -> -2 - 3*I}, {r -> -2 + 3*I}, {r -> 0}, {r -> 0}} ;[o] {{r -> -2 - 3 I}, {r -> -2 + 3 I}, {r -> 0}, {r -> 0}} :[font = text; inactive; preserveAspect] Since r == 0 is repeated, the general soution is ;[s] 3:0,0;6,1;12,0;49,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] y[x_] = c1 + c2 x + E^(-2x)(c3 Cos[3x] + c4 Sin[3x]) :[font = output; output; inactive; preserveAspect; endGroup] c1 + c2*x + (c3*Cos[3*x] + c4*Sin[3*x])/E^(2*x) ;[o] c3 Cos[3 x] + c4 Sin[3 x] c1 + c2 x + ------------------------- 2 x E :[font = text; inactive; preserveAspect] Let's check: :[font = input; preserveAspect; startGroup] L[y[x]] //Simplify :[font = output; output; inactive; preserveAspect; endGroup] 0 ;[o] 0 :[font = subsubsection; inactive; preserveAspect; startGroup] b) Using the method of undetermined coefficients, determine a suitable form for a particular solution of the differential equation L[y] == x^2 + Sin[3x] Do not solve for the constants. ;[s] 4:0,1;2,0;131,2;156,0;189,-1; 3:2,13,9,Times,1,12,0,0,0;1,13,9,Times,3,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = subsubsection; inactive; preserveAspect; endGroup] Solution :[font = text; inactive; preserveAspect] Since r == 0 is a repeated root ofthe characteristic equation, we need to multiply the usual guess of A x^2 + B x + C by x^2 in order to get something like x^2 to result after applying L. To get Sin[3x] we need only try D Cos[3x] + E Sin[3x]. There is no factor of E^(-2x) here so we don't have to worry about multiplying this guess by a power of x. The particular solutionthus has the form: ;[s] 19:0,0;6,1;12,0;102,1;117,0;121,1;124,0;156,1;159,0;185,1;186,0;195,1;202,0;220,1;241,0;265,1;272,0;347,1;348,0;392,-1; 2:10,13,9,Times,0,12,0,0,0;9,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect] yp[x_] = x^2(a x^2 + b x + c) + d Cos[3x] + e Sin[3x]; :[font = text; inactive; preserveAspect] To solve for the constants (this was not required!): :[font = input; preserveAspect; startGroup] L[yp[x]]//Simplify :[font = output; output; inactive; preserveAspect; endGroup] 2*(12*a + 12*b + 13*c + 48*a*x + 39*b*x + 78*a*x^2 - 18*d*Cos[3*x] - 54*e*Cos[3*x] + 54*d*Sin[3*x] - 18*e*Sin[3*x]) ;[o] 2 2 (12 a + 12 b + 13 c + 48 a x + 39 b x + 78 a x - 18 d Cos[3 x] - 54 e Cos[3 x] + 54 d Sin[3 x] - 18 e Sin[3 x]) :[font = input; preserveAspect; startGroup] Collect[%/2, {x,Cos[3x],Sin[3x]}] :[font = output; output; inactive; preserveAspect; endGroup] 12*a + 12*b + 13*c + (48*a + 39*b)*x + 78*a*x^2 + (-18*d - 54*e)*Cos[3*x] + (54*d - 18*e)*Sin[3*x] ;[o] 2 12 a + 12 b + 13 c + (48 a + 39 b) x + 78 a x + (-18 d - 54 e) Cos[3 x] + (54 d - 18 e) Sin[3 x] :[font = input; preserveAspect] eqns = {12a + 12b + 13c == 0, 48a + 39b == 0, 78a == 1, -18d - 54e == 0, 54d - 18e == 1}; :[font = input; preserveAspect; startGroup] Solve[eqns] :[font = output; output; inactive; preserveAspect; endGroup] {{c -> 6/2197, b -> -8/507, d -> 1/60, e -> -1/180, a -> 1/78}} ;[o] 6 8 1 1 1 {{c -> ----, b -> -(---), d -> --, e -> -(---), a -> --}} 2197 507 60 180 78 :[font = input; preserveAspect; startGroup] yp[x_] = yp[x] /. First[%] :[font = output; output; inactive; preserveAspect; endGroup] x^2*(6/2197 - (8*x)/507 + x^2/78) + Cos[3*x]/60 - Sin[3*x]/180 ;[o] 2 2 6 8 x x Cos[3 x] Sin[3 x] x (---- - --- + --) + -------- - -------- 2197 507 78 60 180 ^*)