{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 21 "Chapter 5, Problem 23" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "eqn := diff(diff(x(t),t),t) + (4 + eps*be ta)*x(t) - eps*x(t)^3 - gamma*cos(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG,*-%%diffG6$-%\"xG6#%\"tG-%\"$G6$F,\"\"#\"\"\"*&,&\"\"%F1*&% $epsGF1%%betaGF1F1F1F)F1F1*&F6\"\"\")F)\"\"$F9!\"\"*&%&gammaGF1-%$cosG F+F1F<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "eqn := subs(x(t)= x0(t)+eps*x1(t),eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG,*-%%d iffG6$,&-%#x0G6#%\"tG\"\"\"*&%$epsGF.-%#x1GF,F.F.-%\"$G6$F-\"\"#F.*&,& \"\"%F.*&F0\"\"\"%%betaGF.F.F.F)F.F.*&F0F;)F)\"\"$F;!\"\"*&%&gammaGF.- %$cosGF,F.F@" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "expr := coe ff(eqn,eps,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%exprG,(-%%diffG6$ -%#x0G6#%\"tG-%\"$G6$F,\"\"#\"\"\"F)\"\"%*&%&gammaGF1-%$cosGF+F1!\"\" " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "dsolve(expr = 0,x0(t)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#x0G6#%\"tG,**(%&gammaG\"\"\",& -%$cosG6#,$F'\"\"$#!\"\"\"\"'-F.F&#F3\"\"#F+-F.6#,$F'F7F+F6*(F*\"\"\", &-%$sinGF&#F+F7-F?F/#F+F4F+-F?F9F+F@*&%$_C1GF+F8F " 0 "" {MPLTEXT 1 0 16 "x0(t) := rhs(%):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "x0(t) := combine(subs(\{_C1= a0,_C2=b0\},x0(t)),trig);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%#x0G6# %\"tG,(*&%&gammaG\"\"\"-%$cosGF&F+#F+\"\"$*&%#a0GF+-F-6#,$F'\"\"#F+F+* &%#b0GF+-%$sinGF3F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "ex pr := coeff(eqn,eps,1); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%exprG,* -%%diffG6$-%#x1G6#%\"tG-%\"$G6$F,\"\"#\"\"\"F)\"\"%*&%%betaGF1,(*&%&ga mmaGF1-%$cosGF+F1#F1\"\"$*&%#a0GF1-F96#,$F,F0F1F1*&%#b0GF1-%$sinGF?F1F 1F1F1*$)F5F;\"\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 " expr := combine(expr,trig);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%expr G,Z*()%&gammaG\"\"#\"\"\"%#a0G\"\"\"-%$cosG6#,$%\"tGF)F,#!\"\"\"\"'*(F (F,-F.6#F1F,)%#b0GF)F*#F3F)*(F(F*F6F*)F+F)F*F:*(F+F*F-F*F8F*#!\"$\"\"% *(F*(F'F*F+F*-F.6#,$F1F@F,#F3\"#7-%#x1GF7F@* &)F+\"\"$F*-F.FDF,#F3F@*(F(F*F*(% %betaGF,F(F*F6F*#F,FP*(F'F*F9F*FXF*F2**F(F*F+F*F9F*-FCFUF,F:**F(F*F+F* F9F*-FC6#,$F1\"\"&F,F:*&)F(FPF*FTF*#F3\"$3\"*&F_oF*F6F*#F3\"#O*&FOF*F- F*F>*&)F9FPF*FBF*#F,F@*&FgoF*FXF*F>*(F'F*F9F*-FCFHF,FJ*(F(F*F " 0 "" {MPLTEXT 1 0 29 "eqn1 := coeff(expr,cos(2*t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G,**&%%betaG\"\"\"%#a0GF(F(*&)%&gammaG\"\"#\"\" \"F)F.#!\"\"\"\"'*&F)F.)%#b0GF-F.#!\"$\"\"%*$)F)\"\"$F.F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eqn2 := coeff(expr,sin(2*t));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G,**&%%betaG\"\"\"%#b0GF(F(*&)% #a0G\"\"#\"\"\"F)F.#!\"$\"\"%*&)%&gammaGF-F.F)F.#!\"\"\"\"'*$)F)\"\"$F .F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "So the secular terms are a 0 (beta - gamma^2/6 - 3/4 r0^2) = 0" }}{PARA 0 "" 0 "" {TEXT -1 74 " \+ b0 (beta - gamma^2/6 - 3/4 r0^2) = \+ 0," }}{PARA 0 "" 0 "" {TEXT -1 11 "as desired." }{MPLTEXT 1 0 0 "" }}} }{MARK "10 0 0" 77 }{VIEWOPTS 1 1 0 1 1 1803 }