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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }{TEXT 256 94 "Here's an example of using maple for perturbati
on expansions.  The following is problem 2 (i)." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 257 91 "Since I only wanted to go out to the seco
nd harmonic (the first frequency beyond the lowest" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 258 66 "frequency) I really only needed to take x
(t) = x0(t) + eps*x1(t). " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "I first define the ODE in expr.  N
ote that I have pulled the forcing over onto the left-hand side." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 71 "restart; expr := diff(diff(x(t),t),t) + 1/4*x(t) + eps*x(t)^3 - \+
cos(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%exprG,*-%%diffG6$-%\"xG6
#%\"tG-%\"$G6$F,\"\"#\"\"\"F)#F1\"\"%*&%$epsGF1)F)\"\"$\"\"\"F1-%$cosG
F+!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "expr := subs(x(t
) = x0(t)+eps*x1(t),expr);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%exprG
,,-%%diffG6$,&-%#x0G6#%\"tG\"\"\"*&%$epsGF.-%#x1GF,F.F.-%\"$G6$F-\"\"#
F.F*#F.\"\"%F/F7*&F0\"\"\")F)\"\"$F:F.-%$cosGF,!\"\"" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 23 "Find the O(1) equation:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 18 "coeff(expr,eps,0);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,(-%%diffG6$-%#x0G6#%\"tG-%\"$G6$F*\"\"#\"\"\"F'#F/\"\"
%-%$cosGF)!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "dsolve(%
=0,x0(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#x0G6#%\"tG,**&,&-%$s
inG6#,$F'#\"\"\"\"\"#F1-F,6#,$F'#\"\"$F1#F1F6F0F+F0F0*&,&-%$cosGF3F7-F
;F-!\"#F0F<F0F0*&%$_C1GF0F+\"\"\"F0*&%$_C2GF0F<F@F0" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 104 "Here's something annoying about Maple --- you \+
have to define x0(t) to be the right hand side even though" }}{PARA 0 
"" 0 "" {TEXT -1 23 "you just solved for it." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 16 "x0(t) := rhs(%);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>-%#x0G6#%\"tG,**&,&-%$sinG6#,$F'#\"\"\"\"\"#F1-F,6#,$F'#\"\"$F1
#F1F6F0F+F0F0*&,&-%$cosGF3F7-F;F-!\"#F0F<F0F0*&%$_C1GF0F+\"\"\"F0*&%$_
C2GF0F<F@F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "x0(t) := com
bine(x0(t),trig);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%#x0G6#%\"tG,(-
%$cosGF&#!\"%\"\"$*&%$_C1G\"\"\"-%$sinG6#,$F'#F0\"\"#F0F0*&%$_C2GF0-F*
F3F0F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "x0 should have period 2
Pi" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "x0(t) := subs(\{_C1=0
,_C2=0\},x0(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%#x0G6#%\"tG,$-%
$cosGF&#!\"%\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "expr;" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&%$epsG\"\"\"-%%diffG6$-%#x1G6#%
\"tG-%\"$G6$F-\"\"#F&F&*&F%\"\"\"F*F&#F&\"\"%*&F%F3),&-%$cosGF,#!\"%\"
\"$F2F&F=F3F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "coeff(expr
,eps,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 29 "Now find the O(eps) equation." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 18 "coeff(expr,eps,1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,(-%%diffG6$-%#x1G6#%\"tG-%\"$G6$F*\"\"#\"\"\"F'#F/\"\"
%*$)-%$cosGF)\"\"$\"\"\"#!#k\"#F" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "dsolve(%=0,x1(t));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6
#/-%#x1G6#%\"tG,**&,*-%$sinG6#,$F'#\"\"&\"\"##\"#K\"$N\"-F,6#,$F'#\"\"
(F1#F3\"$*=-F,6#,$F'#\"\"\"F1#F3\"\"*-F,6#,$F'#\"\"$F1#F3\"#FF@F<F@F@*
&,*-%$cosGF6F:-FMF-#!#KF4-FMFDFH-FMF=#FPFBF@FRF@F@*&%$_C1GF@F<\"\"\"F@
*&%$_C2GF@FRFVF@" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "x1(t) :
= rhs(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>-%#x1G6#%\"tG,**&,*-%$si
nG6#,$F'#\"\"&\"\"##\"#K\"$N\"-F,6#,$F'#\"\"(F1#F3\"$*=-F,6#,$F'#\"\"
\"F1#F3\"\"*-F,6#,$F'#\"\"$F1#F3\"#FF@F<F@F@*&,*-%$cosGF6F:-FMF-#!#KF4
-FMFDFH-FMF=#FPFBF@FRF@F@*&%$_C1GF@F<\"\"\"F@*&%$_C2GF@FRFVF@" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "x1(t) := combine(x1(t),trig)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%#x1G6#%\"tG,*-%$cosG6#,$F'\"\"
$#!#k\"$X*-F*F&#F/\"#F*&%$_C1G\"\"\"-%$sinG6#,$F'#F6\"\"#F6F6*&%$_C2GF
6-F*F9F6F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "x1(t) := subs
(\{_C1=0,_C2=0\},x1(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%#x1G6#%
\"tG,&-%$cosG6#,$F'\"\"$#!#k\"$X*-F*F&#F/\"#F" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 26 "x(t) := x0(t) + eps*x1(t);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>-%\"xG6#%\"tG,&-%$cosGF&#!\"%\"\"$*&%$epsG\"\"\",&-F*6
#,$F'F-#!#k\"$X*F)#F6\"#FF0F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "
In this problem, epsilon = -1/2, so I substitute in at the last minute
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "x(t) := subs(eps=-1/2,x
(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"xG6#%\"tG,&-%$cosGF&#!\"
%\"#F-F*6#,$F'\"\"$#\"#K\"$X*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}}{MARK "13 0 0" 29 }{VIEWOPTS 1 1 0 1 1 1803 }