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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 24 "ENGI 3424 Example 5.11.3
" }}{PARA 0 "" 0 "" {TEXT -1 72 "Series Solution of a Second Order Lin
ear ODE (non-constant coefficients)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "with(DEtools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 62 "ode := (1-x^2)*diff(y(x),x,x) - 5*x*diff(y(x),x) - 3*y(x) = 0;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG/,(*&,&\"\"\"F)*$)%\"xG\"\"
#F)!\"\"F)-%%diffG6$-%\"yG6#F,-%\"$G6$F,F-F)F)*(\"\"&F)F,F)-F06$F2F,F)
F.*&\"\"$F)F2F)F.\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "O
rder := 11;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&OrderG\"#6" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "dsolve(ode, y(x), series);" 
}}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG+;F'-F%6#\"\"!F+--%\"D
G6#F%F*\"\"\",$*&#\"\"$\"\"#F0F)F0F0F5,$*&#\"\"%F4F0F,F0F0F4,$*&#\"#:
\"\")F0F)F0F0F9,$*&#F>\"\"&F0F,F0F0FB,$*&#\"#N\"#;F0F)F0F0\"\"',$*&#\"
#kFFF0F,F0F0\"\"(,$*&#\"$:$\"$G\"F0F)F0F0F>,$*&#FR\"#jF0F,F0F0\"\"*,$*
&#\"$$p\"$c#F0F)F0F0\"#5-%\"OG6#F0\"#6" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 98 "In order to see the parts of the general solution more cl
early, apply a set of initial conditions." }}{PARA 0 "" 0 "" {TEXT -1 
38 "The part of the solution multiplying  " }{TEXT 257 2 "y'" }{TEXT 
-1 7 "(0) is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "ics := y(0
)=0, D(y)(0)=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$icsG6$/-%\"yG6#
\"\"!F*/--%\"DG6#F(F)\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
33 "dsolve(\{ode, ics\}, y(x), series);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%\"yG6#%\"xG+/F'\"\"\"F)#\"\"%\"\"$F,#\"\")\"\"&F/#\"#k\"#N\"
\"(#\"$G\"\"#j\"\"*-%\"OG6#F)\"#6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
38 "The part of the solution multiplying  " }{TEXT 258 1 "y" }{TEXT 
-1 7 "(0) is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "ics := y(0
)=1, D(y)(0)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$icsG6$/-%\"yG6#
\"\"!\"\"\"/--%\"DG6#F(F)F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
33 "dsolve(\{ode, ics\}, y(x), series);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%\"yG6#%\"xG+1F'\"\"\"\"\"!#\"\"$\"\"#F-#\"#:\"\")\"\"%#\"#N\"
#;\"\"'#\"$:$\"$G\"F0#\"$$p\"$c#\"#5-%\"OG6#F)\"#6" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 68 "The exact solution requires methods beyond the sc
ope of this course:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "dsol
ve(ode, y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*,,&
\"\"\"!\"\"F'F+F,,&F'F+F+F+F,,&-%#lnG6#,&F'F+*$,&F+F,*$)F'\"\"#F+F+#F+
F7F+F,*&F'F+F4F8F+F+F4#F,F7%$_C1GF+F+*(%$_C2GF+F*#!\"$F7F-F>F+" }}}
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