{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 12" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "eqn := diff(diff(x(t),t),t) + w^2*x(t) - eps*x(t)^3 - gamma*cos(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG,*-%%diffG6$-%\"xG6#%\"tG-%\"$G6$F,\"\"#\"\"\"*&)%\"wGF0 \"\"\"F)F1F1*&%$epsGF1)F)\"\"$F5!\"\"*&%&gammaGF1-%$cosGF+F1F:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "eqn := subs(x(t) = x1(t) + x 3(t),eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG,*-%%diffG6$,&-%# x1G6#%\"tG\"\"\"-%#x3GF,F.-%\"$G6$F-\"\"#F.*&)%\"wGF4\"\"\"F)F.F.*&%$e psGF.)F)\"\"$F8!\"\"*&%&gammaGF.-%$cosGF,F.F=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "eqn := subs(\{x1(t) = a*cos(t),x3(t)=b*cos(3*t) \},eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG,,-%%diffG6$*&%\"aG \"\"\"-%$cosG6#%\"tGF+-%\"$G6$F/\"\"#F+-F'6$*&%\"bGF+-F-6#,$F/\"\"$F+F 0F+*&)%\"wGF3\"\"\",&F)F+F6F+F+F+*&%$epsGF+)F@F;F?!\"\"*&%&gammaGF+F,F ?FD" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "eqn := combine(eqn,t rig);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$eqnG,@*&%&gammaG\"\"\"-%$c osG6#%\"tGF(!\"\"*(%$epsGF()%\"bG\"\"$\"\"\"-F*6#,$F,\"\"*F(#F-\"\"%*( F/F3)%\"aGF2F3-F*6#,$F,F2F(F8*(F/F3F;F3F)F3#!\"$F9*&F " 0 "" {MPLTEXT 1 0 25 "eqn1:= coeff(eqn, cos(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G,.%&gammaG!\"\"*&% $epsG\"\"\")%\"aG\"\"$\"\"\"#!\"$\"\"%F,F'*&)%\"wG\"\"#F.F,F*F**(F)F.) F,F5F.%\"bGF*F/*(F)F.F,F.)F8F5F.#F0F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "eqn2 := coeff(eqn,cos(3*t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G,,*&%$epsG\"\"\")%\"aG\"\"$\"\"\"#!\"\"\"\"%%\" bG!\"**&)%\"wG\"\"#F,F0F(F(*(F'F,)F*F5F,F0F,#!\"$F5*&F'F,)F0F+F,#F9F/ " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "So a and b must satisfy the a bove equations. Let's now assume a=a0+eps*a1, b = b0+eps*b1 and" }} {PARA 0 "" 0 "" {TEXT -1 28 "at least find a0,a1,b0,b1..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "a := a0+eps*a1; b := b0+eps*b1;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG,&%#a0G\"\"\"*&%$epsGF'%#a1GF'F' " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG,&%#b0G\"\"\"*&%$epsGF'%#b1G F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "coeff(eqn1,eps,0); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(%&gammaG!\"\"%#a0GF%*&)%\"wG\"\" #\"\"\"F&\"\"\"F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "coeff( eqn2,eps,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%#b0G!\"**&)%\"wG\" \"#\"\"\"F$\"\"\"F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "solv e(coeff(eqn1,eps,0)=0,a0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%&gamm aG\"\"\",&!\"\"\"\"\"*$)%\"wG\"\"#F%F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "a0 := %;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a0G* &%&gammaG\"\"\",&!\"\"\"\"\"*$)%\"wG\"\"#F'F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "solve(coeff(eqn2,eps,0)=0,b0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "b0 := 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b0G\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "coeff(eqn1,eps,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&*$)%&gammaG\"\"$\"\"\"F)*$),&!\"\"\"\"\" *$)%\"wG\"\"#F)F.\"\"$F)!\"\"#!\"$\"\"%%#a1GF-*&F0F)F8F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "coeff(eqn2,eps,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&*$)%&gammaG\"\"$\"\"\"F)*$),&!\"\"\"\"\"*$)%\" wG\"\"#F)F.\"\"$F)!\"\"#F-\"\"%%#b1G!\"**&F0F)F7F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "solve(coeff(eqn1,eps,1)=0,a1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*$)%&gammaG\"\"$\"\"\"F),,\"\"\"F+*$)%\" wG\"\"#F)!\"%*$)F.\"\"%F)\"\"'*$)F.F4F)F0*$)F.\"\")F)F+!\"\"#F(F3" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "a1 := factor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a1G,$*&*$)%&gammaG\"\"$\"\"\"F+*&),&%\"wG \"\"\"!\"\"F0\"\"%F+),&F/F0F0F0\"\"%F+!\"\"#F*\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "solve(coeff(eqn2,eps,1)=0,b1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*$)%&gammaG\"\"$\"\"\"F),,\"\"*\"\"\"*$) %\"wG\"\"#F)!#G*$)F/\"\"%F)\"#I*$)F/\"\"'F)!#7*$)F/\"\")F)F,!\"\"#F,F4 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "b1 := factor(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b1G,$*&*$)%&gammaG\"\"$\"\"\"F+**,& %\"wG\"\"\"!\"$F/\"\"\",&F.F/F*F/\"\"\"),&F.F/!\"\"F/\"\"$F+),&F.F/F/F /\"\"$F+!\"\"#F/\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "x( t) = a*cos(t)+b*cos(3*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6# %\"tG,&*&,&*&%&gammaG\"\"\",&!\"\"\"\"\"*$)%\"wG\"\"#F-F0!\"\"F0*&*&%$ epsGF0)F,\"\"$F-F-*&),&F3F0F/F0\"\"%F-),&F3F0F0F0\"\"%F-F5#F:\"\"%F0-% $cosGF&F0F0*&*(F8F-F9F--FE6#,$F'F:F0F-**,&F3F0!\"$F0\"\"\",&F3F0F:F0\" \"\")F=\"\"$F-)F@\"\"$F-F5#F0FC" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "22 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }