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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 14 "Example 1.10.1" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 57 "ode := diff(y(x),x,x) - x*diff(y(x),x) + exp(x)*y(x) \+
= 4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG/,(-%%diffG6$-%\"yG6#%
\"xG-%\"$G6$F-\"\"#\"\"\"*&F-F2-F(6$F*F-F2!\"\"*&-%$expGF,F2F*F2F2\"\"
%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "ics := y(0) = 1, D(y)(
0) = 4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$icsG6$/-%\"yG6#\"\"!\"\"
\"/--%\"DG6#F(F)\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "ds
olve(\{ode,ics\}, y(x), series);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-
%\"yG6#%\"xG+1F'\"\"\"\"\"!\"\"%F)#\"\"$\"\"#F.#!\"\"\"\"'F-#F0F+F+#F0
\"\"&F4-%\"OG6#F)F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Example 1.
10.2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "ode := diff(y(x),x,
x) + (x^2)*y(x) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG/,&-%%d
iffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&)F-F1F2F*F2F2\"\"!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Order := 16;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%&OrderG\"#;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 47 "dsolve(\{ode, y(0)=A, D(y)(0)=B\}, y(x), series);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG+5F'%\"AG\"\"!%\"BG\"\"
\",$*&\"#7!\"\"F)F,F0\"\"%,$*&\"#?F0F+F,F0\"\"&,$*&\"$s'F0F)F,F,\"\"),
$*&\"%S9F0F+F,F,\"\"*,$*&\"&/())F0F)F,F0F/,$*&\"'SYAF0F+F,F0\"#8-%\"OG
6#F,\"#;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Example 1.12.1   Bess
el's ODE" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Order := 8:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "ode := x^2*diff(y(x),x,x) +
 x*diff(y(x),x) + (x^2-n^2)*y(x) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%$odeG/,(*&)%\"xG\"\"#\"\"\"-%%diffG6$-%\"yG6#F)-%\"$G6$F)F*F+F+*&
F)F+-F-6$F/F)F+F+*&,&*$F(F+F+*$)%\"nGF*F+!\"\"F+F/F+F+\"\"!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "dsolve(ode, y(x), series);" 
}}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*(%$_C1G\"\"\")F'%\"
nGF++-F'F+\"\"!*&F+F+,&*&\"\"%F+F-F+!\"\"F3F4F4\"\"#*&F+F+*&,&*&\"\")F
+F-F+F4\"#;F4F+F1F+F4F3*&F+F+*(,&*&\"#7F+F-F+F4\"#OF4F+F8F+F1F+F4\"\"'
-%\"OG6#F+F:F+F+*(%$_C2GF+)F',$F-F4F++-F'F+F/*&F+F+,&*&F3F+F-F+F+F3F4F
4F5*&F+F+*&,&*&F:F+F-F+F+F;F4F+FLF+F4F3*&F+F+*(,&*&F@F+F-F+F+FAF4F+FPF
+FLF+F4FBFCF:F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simpli
fy(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*(%$_C1G\"\"
\")F'%\"nGF++-F'F+\"\"!,$*&F+F+*&\"\"%F+,&F-F+F+F+F+!\"\"F5\"\"#,$*&F+
F+*(\"#KF+,&F-F+F6F+F+F4F+F5F+F3,$*&F+F+**\"$%QF+,&F-F+\"\"$F+F+F;F+F4
F+F5F5\"\"'-%\"OG6#F+\"\")F+F+*(%$_C2GF+)F',$F-F5F++-F'F+F/,$*&F+F+*&F
3F+,&F-F+F+F5F+F5F+F6,$*&F+F+*(F:F+,&F-F+F6F5F+FOF+F5F+F3,$*&F+F+**F?F
+,&F-F+FAF5F+FSF+FOF+F5F+FBFCFFF+F+" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "dsolve(ode, y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#/-%\"yG6#%\"xG,&*&%$_C1G\"\"\"-%(BesselJG6$%\"nGF'F+F+*&%$_C2GF+-%(Be
sselYGF.F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Example 1.12.2   \+
LeGendre's ODE" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "ode := (1
-x^2)*diff(y(x),x,x) - 2*x*diff(y(x),x) + p*(p+1)*y(x) = 0;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%$odeG/,(*&,&\"\"\"F)*$)%\"xG\"\"#F)!\"\"F)
-%%diffG6$-%\"yG6#F,-%\"$G6$F,F-F)F)*(F-F)F,F)-F06$F2F,F)F.*(%\"pGF),&
F<F)F)F)F)F2F)F)\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "ic
s := y(0) = A, D(y)(0) = B;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$icsG
6$/-%\"yG6#\"\"!%\"AG/--%\"DG6#F(F)%\"BG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "dsolve(\{ode, ics\}, y(x), series);" }}{PARA 12 "" 1 
"" {XPPMATH 20 "6#/-%\"yG6#%\"xG+5F'%\"AG\"\"!%\"BG\"\"\",$**\"\"#!\"
\"%\"pGF,,&F1F,F,F,F,F)F,F0F/,(*&#F,\"\"$F,F+F,F,*&#F,\"\"'F,*&)F1F/F,
F+F,F,F0*&#F,F9F,*&F1F,F+F,F,F0F6,**&#\"\"&\"#CF,*&F;F,F)F,F,F0*&#F,\"
\"%F,*&F1F,F)F,F,F0*&#F,FCF,*&)F1FGF,F)F,F,F,*&#F,\"#7F,*&)F1F6F,F)F,F
,F,FG,,*&#\"#8\"$?\"F,F:F,F0*&#\"\"(\"#gF,F>F,F0*&#F,FVF,*&FLF,F+F,F,F
,*&#F,FZF,*&FQF,F+F,F,F,*&#F,FBF,F+F,F,FB,.*&#\"#Z\"$g$F,FDF,F0*&#F,F9
F,FHF,F0*&#\"#B\"$?(F,FKF,F,*&#\"#<\"$S#F,FPF,F,*&#F,FgoF,*&)F1F9F,F)F
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\"F,FjnF,F,*&#FB\"#jF,F:F,F0*&#F,FhpF,*&F_pF,F+F,F,F0*&#F,F\\qF,*&FcpF
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" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*2-%&GAMMAG6#,&#\"
\"\"\"\"#F/*&F0!\"\"%\"pGF/F2F/-F+6#,&F/F/*&F0F2F3F/F/F/-F+6#,$*&F0F2F
3F/F2F/-F+6#,&#\"\"$F0F/*&F0F2F3F/F/F/,(*&-%*LegendreQG6$F3\"\"!F/%\"B
GF/F/*(%\"AGF/-FE6$,&F3F/F/F/FGF/F3F/F/*&FJF/FKF/F/F/%#PiG#F2F0,***FKF
/F8F/F<F/F3F/F2*(FKF/F8F/F<F/F2**FDF/F*F/F4F/F3F/F/*(FDF/F*F/F4F/F/F2-
%*LegendrePG6$F3F'F/F2*(FQF2,(*(FHF/F8F/F<F/F/**FJF/F*F/F4F/F3F/F/*(FJ
F/F*F/F4F/F/F/-FEFXF/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
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