(*^ ::[ 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, noPageBreakBelow, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L1, 12, "Courier"; ; fontset = output, output, inactive, noPageBreakBelow, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; ; fontset = message, inactive, noPageBreakBelow, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = print, inactive, noPageBreakBelow, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = info, inactive, noPageBreakBelow, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakBelow, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, L1, 12, "Courier"; ; fontset = name, inactive, noPageBreakInGroup, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, B65535, L1, 10, "Times"; ; fontset = header, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = leftheader, 12; fontset = footer, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M7, L1, 12; fontset = leftfooter, 12; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 10, "Times"; ; 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] HERBERT SPENCER 1820Ð1903 Evolution¼isÐÐa change from an indefinite, incoherent homogeneity, to a definite coherent heterogeneity. First Principles (1862), ch.16, 138 ;[s] 3:0,0;131,1;148,2;167,-1; 3:1,11,8,Times,0,12,0,0,0;1,9,7,Times,2,10,0,0,0;1,9,7,Times,0,10,0,0,0; :[font = title; inactive; preserveAspect; startGroup] Nonhomogeneous Equations, Undetermined Coefficients :[font = subtitle; inactive; preserveAspect] Steve Dunbar Department of Mathematics and Statistics University of Nebraska-Lincoln Lincoln, NE 68588-0323 sdunbar@mathlab01.unl.edu Math 221H, Fall 1993 :[font = subsubtitle; inactive; preserveAspect] Covers Nagle & Saff, Sections 4.8, 6.4 :[font = section; inactive; Cclosed; preserveAspect; startGroup] The Method of Undetermined Coefficients: Example 1 :[font = text; inactive; preserveAspect] The goal is to solve a non-homogeneous differential equation y'' + 25 y = 10 cos(7 t) y(0) = 0 y'(0) = 0 with the method of undetermined coefficients: :[font = subsection; inactive; preserveAspect; startGroup] Step 1: Find the solution of the corresponding homogeneous equation :[font = input; preserveAspect] hsoln = DSolve[ y''[t] + 25 y[t] == 0, y[t], t] :[font = input; preserveAspect; endGroup] yh = First[ y[t] /. hsoln] :[font = subsection; inactive; preserveAspect; startGroup] Step 2: Apply annihilator and solve expanded equation :[font = input; preserveAspect] L[f_] := Expand[ D[f,{t,2}] + 49*f] :[font = input; preserveAspect] bigeqn = L[y''[t] + 25*y[t]] == L[Sin[7*t] ] :[font = input; preserveAspect; endGroup] DSolve[ bigeqn, y[t], t] :[font = subsection; inactive; preserveAspect; startGroup] Step 3: Identify particular solution and find undetermined coefficients :[font = input; preserveAspect] yp = A Cos[7 t] + B Sin[ 7 t] :[font = input; preserveAspect] D[yp, {t, 2}] + 25 yp :[font = input; preserveAspect] Simplify[%] :[font = input; preserveAspect] uceqns = {-24 A == 10, -24 B == 0} :[font = input; preserveAspect] Solve[uceqns, {A,B}] :[font = input; preserveAspect] y = yh + yp /. % :[font = input; preserveAspect; endGroup] y = First[y] :[font = subsection; inactive; preserveAspect; startGroup] Determine arbitrary coefficients from initial conditions :[font = input; preserveAspect] ic1 = ( y /. t -> 0) == 0 :[font = input; preserveAspect] ic2 = ( D[y, t] /. t -> 0) == 0 :[font = input; preserveAspect] Solve[ {ic1, ic2}, {C[1], C[2]}] :[font = input; preserveAspect] y = y /. % :[font = input; preserveAspect] y = First[%] :[font = input; preserveAspect; startGroup] Plot[ y, {t, 0, 10}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup] The Unformatted text for this cell was not generated. Use options in the Actions Settings dialog box to control when Unformatted text is generated. ;[o] -Graphics- :[font = section; inactive; Cclosed; preserveAspect; startGroup] The Method of Undetermined Coefficients: Example 2 :[font = text; inactive; preserveAspect] The goal is to solve a very similar non-homogeneous differential equation y'' + 25 y = 10 cos(5 t) y(0) = 0 y'(0) = 0 with the method of undetermined coefficients: The results will be very different and the differences are important! :[font = input; preserveAspect] Clear[ y,t, hsoln, L, f, bigeqn, yp, A, B,] :[font = subsection; inactive; preserveAspect; startGroup] Step 1: Find the solution of the corresponding homogeneous equation :[font = input; preserveAspect] hsoln = DSolve[ y''[t] + 25 y[t] == 0, y[t], t] :[font = input; preserveAspect; endGroup] yh = First[ y[t] /. hsoln] :[font = subsection; inactive; preserveAspect; startGroup] Step 2: Apply annihilator and solve expanded equation :[font = input; preserveAspect] L[f_] := Expand[ D[f,{t,2}] + 25*f] :[font = input; preserveAspect] bigeqn = L[y''[t] + 25*y[t]] == L[Sin[5*t] ] :[font = input; preserveAspect] Solve[ r^4 + 50*r^2 + 625 == 0, r] :[font = input; preserveAspect; endGroup] DSolve[ bigeqn, y[t], t] :[font = subsection; inactive; preserveAspect; startGroup] Step 3: Identify particular solution and find undetermined coefficients :[font = input; preserveAspect] yp = A*t*Cos[5 t] + B*t*Sin[ 5 t] :[font = input; preserveAspect] D[yp, {t, 2}] + 25 yp :[font = input; preserveAspect] Simplify[%] :[font = input; preserveAspect] uceqns = {-10 A == 0, 10 B == 10} :[font = input; preserveAspect] Solve[uceqns, {A,B}] :[font = input; preserveAspect] y = yh + yp /. % :[font = input; preserveAspect] y = First[y] :[font = input; preserveAspect; endGroup] Simplify[ D[y, {t,2}] + 25*y] :[font = subsection; inactive; preserveAspect; startGroup] Determine arbitrary coefficients from initial conditions :[font = input; preserveAspect] ic1 = ( y /. t -> 0) == 0 :[font = input; preserveAspect] ic2 = ( D[y, t] /. t -> 0) == 0 :[font = input; preserveAspect] Solve[ {ic1, ic2}, {C[1], C[2]}] :[font = input; preserveAspect] y = y /. % :[font = input; preserveAspect] y = First[%] :[font = input; preserveAspect; endGroup; endGroup] Plot[ y, {t, 0, 10}] :[font = section; inactive; Cclosed; preserveAspect; startGroup] Exercise :[font = text; inactive; preserveAspect; endGroup; endGroup] Solve the non-homogeneous, second-order, initial value problem: y'' + 25 y = t cos(5 t) y(0) = 0 y'(0) = 0 Notice that because of the structure of the right side, the annihilator will have to be (D2 + 25)2 ;[s] 4:0,0;206,1;207,2;213,3;214,-1; 4:1,11,8,Times,0,12,0,0,0;1,11,8,Times,32,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,11,8,Times,32,12,0,0,0; ^*)