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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 40 "Finding the equations to \+
make figure 7.4" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "restart : eqn := diff(diff
(x(t),t),t) + k*diff(x(t),t) + x(t) + beta*x(t)^3 - g*cos(w*t);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG,,-%%diffG6$-%\"xG6#%\"tG-%\"$G
6$F,\"\"#\"\"\"*&%\"kGF1-F'6$F)F,F1F1F)F1*&%%betaGF1)F)\"\"$\"\"\"F1*&
%\"gGF1-%$cosG6#*&%\"wGF1F,F1F1!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 52 "eqn := subs(\{x(t)=a(t)*cos(w*t)+b(t)*sin(w*t)\},eqn)
;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$eqnG,.-%%diffG6$,&*&-%\"aG6#%
\"tG\"\"\"-%$cosG6#*&%\"wGF/F.F/F/F/*&-%\"bGF-F/-%$sinGF2F/F/-%\"$G6$F
.\"\"#F/*&%\"kGF/-F'6$F)F.F/F/F*F/F5F/*&%%betaGF/)F)\"\"$\"\"\"F/*&%\"
gGF/F0FF!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "eqn := sub
s(\{diff(diff(a(t),t),t)=0,diff(diff(b(t),t),t)=0\},eqn);" }}{PARA 12 
"" 1 "" {XPPMATH 20 "6#>%$eqnG,4*(-%%diffG6$-%\"aG6#%\"tGF-\"\"\"-%$si
nG6#*&%\"wGF.F-F.F.F3\"\"\"!\"#*(F*F.-%$cosGF1F.)F3\"\"#F4!\"\"*(-F(6$
-%\"bGF,F-F.F7F4F3F4F:*(F?F.F/F4F9F4F;*&%\"kGF.,**&F'F4F7F4F.*(F*F4F/F
4F3F4F;*&F=F4F/F4F.*(F?F4F7F4F3F4F.F.F.*&F*F4F7F4F.*&F?F4F/F4F.*&%%bet
aGF.),&FIF.FJF.\"\"$F4F.*&%\"gGF.F7F4F;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "eqn := combine(eqn,trig);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%$eqnG,H*(-%%diffG6$-%\"aG6#%\"tGF-\"\"\"-%$sinG6#*&%
\"wGF.F-F.F.F3\"\"\"!\"#*(F*F.-%$cosGF1F.)F3\"\"#F4!\"\"*(-F(6$-%\"bGF
,F-F.F7F4F3F4F:*(F?F.F/F4F9F4F;*(%\"kGF.F'F4F7F4F.**FCF4F*F4F/F4F3F4F;
*(FCF4F=F4F/F4F.**FCF4F?F4F7F4F3F4F.*&F*F4F7F4F.*&F?F4F/F4F.*(%%betaGF
.)F*\"\"$F4-F86#,$F2FLF.#F.\"\"%*(FJF4FKF4F7F4#FLFQ**FJF4)F*F:F4F?F4-F
0FNF.FS**FJF4FUF4F?F4F/F4FS**FJF4F*F4F7F4)F?F:F4FS**FJF4F*F4FYF4FMF4#!
\"$FQ*(FJF4)F?FLF4FVF4#F;FQ*(FJF4FhnF4F/F4FS*&%\"gGF.F7F4F;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "eqn1 := coeff(eqn,cos(w*t));
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G,2*&-%\"aG6#%\"tG\"\"\")%
\"wG\"\"#\"\"\"!\"\"*&-%%diffG6$-%\"bGF)F*F+F-F+F.*&%\"kGF+-F36$F'F*F+
F+*(F8F/F5F+F-F/F+F'F+*&%%betaGF+)F'\"\"$F/#F?\"\"%*(F=F/F'F/)F5F.F/F@
%\"gGF0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "eqn2 := coeff(eq
n,sin(w*t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G,0*&-%%diffG6$
-%\"aG6#%\"tGF-\"\"\"%\"wGF.!\"#*&-%\"bGF,F.)F/\"\"#\"\"\"!\"\"*(%\"kG
F.F*F.F/F6F7*&F9F6-F(6$F2F-F.F.F2F.*(%%betaGF.)F*F5F6F2F6#\"\"$\"\"%*&
F>F6)F2FAF6F@" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "sols := so
lve(\{eqn1=0,eqn2=0\},\{diff(a(t),t),diff(b(t),t)\});" }}{PARA 12 "" 
1 "" {XPPMATH 20 "6#>%%solsG<$/-%%diffG6$-%\"bG6#%\"tGF-,$*&,6*(%\"kG
\"\"\"F*F3)%\"wG\"\"#\"\"\"\"\"%*()F2F6F7-%\"aGF,F3F5F3!\"%*&F*F7F2F7F
8**F2F7%%betaGF3)F;F6F7F*F7\"\"$*(F2F7F@F7)F*FBF7FB*&F;F7)F5FBF7!\")*&
F5F7F;F7\"\")*(F@F7)F;FBF7F5F7\"\"'**F@F7F;F7)F*F6F7F5F7FL*&%\"gGF3F5F
7FGF7,&*$F:F7F3*$F4F7F8!\"\"#!\"\"F8/-F(6$F;F-,$*&,6*(F*F7F5F7F:F7F=*&
F*F7FFF7FG*(F2F7F;F7F4F7F=*&F2F7F;F7F=*(F2F7F@F7FKF7!\"$**F2F7F@F7F;F7
FNF7F\\o*&F2F7FPF7F8*&F*F7F5F7FI**F@F7FAF7F*F7F5F7FL*(F@F7FDF7F5F7FLF7
FQFT#F3F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "sols[1];" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"bG6#%\"tGF*,$*&,6*(%\"k
G\"\"\"F'F0)%\"wG\"\"#\"\"\"\"\"%*()F/F3F4-%\"aGF)F0F2F0!\"%*&F'F4F/F4
F5**F/F4%%betaGF0)F8F3F4F'F4\"\"$*(F/F4F=F4)F'F?F4F?*&F8F4)F2F?F4!\")*
&F2F4F8F4\"\")*(F=F4)F8F?F4F2F4\"\"'**F=F4F8F4)F'F3F4F2F4FI*&%\"gGF0F2
F4FDF4,&*$F7F4F0*$F1F4F5!\"\"#!\"\"F5" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "RHS2 := rhs(sols[1]);" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#>%%RHS2G,$*&,6*(%\"kG\"\"\"-%\"bG6#%\"tGF*)%\"wG\"\"#\"\"\"\"\"%
*()F)F1F2-%\"aGF-F*F0F*!\"%*&F+F2F)F2F3**F)F2%%betaGF*)F6F1F2F+F2\"\"$
*(F)F2F;F2)F+F=F2F=*&F6F2)F0F=F2!\")*&F0F2F6F2\"\")*(F;F2)F6F=F2F0F2\"
\"'**F;F2F6F2)F+F1F2F0F2FG*&%\"gGF*F0F2FBF2,&*$F5F2F**$F/F2F3!\"\"#!\"
\"F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "RHS1 := rhs(sols[2]
);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%RHS1G,$*&,6*(-%\"bG6#%\"tG\"
\"\"%\"wGF-)%\"kG\"\"#\"\"\"!\"%*&F)F2)F.\"\"$F2!\")*(F0F--%\"aGF+F-)F
.F1F2F3*&F0F2F9F2F3*(F0F2%%betaGF-)F9F6F2!\"$**F0F2F>F2F9F2)F)F1F2F@*&
F0F2%\"gGF-\"\"%*&F)F2F.F2\"\")**F>F2)F9F1F2F)F2F.F2\"\"'*(F>F2)F)F6F2
F.F2FJF2,&*$F/F2F-*$F;F2FE!\"\"#F-FE" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "RHS1 := subs(\{a(t)=a,b(t)=b\},RHS1);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%%RHS1G,$*&,6*(%\"bG\"\"\"%\"wGF*)%\"kG\"\"#\"\"
\"!\"%*&F)F/)F+\"\"$F/!\")*(F-F*%\"aGF*)F+F.F/F0*&F-F/F6F/F0*(F-F/%%be
taGF*)F6F3F/!\"$**F-F/F:F/F6F/)F)F.F/F<*&F-F/%\"gGF*\"\"%*&F)F/F+F/\"
\")**F:F/)F6F.F/F)F/F+F/\"\"'*(F:F/)F)F3F/F+F/FFF/,&*$F,F/F**$F7F/FA!
\"\"#F*FA" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "RHS2 := subs(
\{a(t)=a,b(t)=b\},RHS2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%RHS2G,$
*&,6*(%\"kG\"\"\"%\"bGF*)%\"wG\"\"#\"\"\"\"\"%*()F)F.F/%\"aGF*F-F*!\"%
*&F+F/F)F/F0**F)F/%%betaGF*)F3F.F/F+F/\"\"$*(F)F/F7F/)F+F9F/F9*&F3F/)F
-F9F/!\")*&F-F/F3F/\"\")*(F7F/)F3F9F/F-F/\"\"'**F7F/F3F/)F+F.F/F-F/FC*
&%\"gGF*F-F/F>F/,&*$F2F/F**$F,F/F0!\"\"#!\"\"F0" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 12 "numer(RHS1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#,6*(%\"bG\"\"\"%\"wGF&)%\"kG\"\"#\"\"\"!\"%*&F%F+)F'\"\"$F+!\")*(F)F
&%\"aGF&)F'F*F+F,*&F)F+F2F+F,*(F)F+%%betaGF&)F2F/F+!\"$**F)F+F6F+F2F+)
F%F*F+F8*&F)F+%\"gGF&\"\"%*&F%F+F'F+\"\")**F6F+)F2F*F+F%F+F'F+\"\"'*(F
6F+)F%F/F+F'F+FB" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "denom(R
HS1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%\"kG\"\"#\"\"\"\"\"%*$)
%\"wGF'F(\"#;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "numer(RHS2
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,6*(%\"kG\"\"\"%\"bGF&)%\"wG\"\"
#\"\"\"!\"%*()F%F*F+%\"aGF&F)F&\"\"%*&F'F+F%F+F,**F%F+%%betaGF&)F/F*F+
F'F+!\"$*(F%F+F3F+)F'\"\"$F+F5*&F/F+)F)F8F+\"\")*&F)F+F/F+!\")*(F3F+)F
/F8F+F)F+!\"'**F3F+F/F+)F'F*F+F)F+F@*&%\"gGF&F)F+F;" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 12 "denom(RHS2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&*$)%\"kG\"\"#\"\"\"\"\"%*$)%\"wGF'F(\"#;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "g := .005; beta := -.167; w := .975
 ; k := .005;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG$\"\"&!\"$" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG$!$n\"!\"$" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"wG$\"$v*!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%\"kG$\"\"&!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "solve(\{
RHS1=0,RHS2=0\},\{a,b\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%<$/%\"bG$
\"+5#zd/\"!#6/%\"aG$\"+jwOI5!#5<$/F%$\"+4!Q:H$F-/F*$\"+\\1-)y%F-<$/F*$
!+wiSf]F-/F%$\"+@)45H%F-" }}}}{MARK "19" 0 }{VIEWOPTS 1 1 0 1 1 1803 }