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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 24 "ENGI 3424 Example 5.11.1
" }}{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 56 "ode := diff(y(x),x,x) - x*diff(y(x),x) + (e^x)*y(x) =
 4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG/,(-%%diffG6$-%\"yG6#%\"
xG-%\"$G6$F-\"\"#\"\"\"*&F-F2-F(6$F*F-F2!\"\"*&)%\"eGF-F2F*F2F2\"\"%" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Order := 11;" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%&OrderG\"#6" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "dsolve(ode, y(x), series);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#/-%\"yG6#%\"xG+;F'-F%6#\"\"!F+--%\"DG6#F%F*\"\"\",&\"\"
#F0*&#F0F2F0F)F0!\"\"F2,$*&#F0\"\"'F0*&-%#lnG6#%\"eGF0F)F0F0F5\"\"$,*#
F0F9F0*&#F0\"#CF0F)F0F5*&#F0\"#7F0*&F;F0F,F0F0F5*&#F0FDF0*&)F;F2F0F)F0
F0F5\"\"%,**&#F0\"$?\"F0F:F0F0*&#F0\"#5F0F;F0F5*&#F0\"#SF0*&FLF0F,F0F0
F5*&#F0FQF0*&)F;F?F0F)F0F0F5\"\"&,0#F0\"#gF0*&#F0\"$S#F0F)F0F5*&#F0FQF
0FHF0F5*&#\"\"(\"$?(F0FKF0F0*&#F0\"#IF0*$FLF0F0F5*&#F0\"$!=F0*&FfnF0F,
F0F0F5*&#F0FcoF0*&)F;FMF0F)F0F0F5F9,0*&#F0\"$g&F0F:F0F0*&#\"#<\"%g7F0F
;F0F5*&#F0\"%?DF0FXF0F5*&#F0F]oF0FenF0F0*&#F0\"$E\"F0*$FfnF0F0F5*&#F0
\"%S]F0*&)F;FgnF0F)F0F0F5*&#F0\"%35F0*&F_pF0F,F0F0F5Fbo,6#F0\"$s'F0*&#
F0\"%)o#F0F)F0F5*&#F0\"%W8F0FHF0F5*&#\"#6\"&!35F0FKF0F0*&#F?\"%?6F0Fgo
F0F5*&#F0\"%S9F0F[pF0F0*&#Ffp\"&SM\"F0F^pF0F0*&#F0F\\rF0*$F_pF0F0F5*&#
F0\"&?.%F0*&)F;F9F0F)F0F0F5*&#F0\"%?nF0*&FeqF0F,F0F0F5\"\"),6*&#Fgn\"&
#>CF0F:F0F0*&#\"#T\"&S-$F0F;F0F5*&#F0\"&'47F0FXF0F0*&#\"#8\"&S=&F0FenF
0F0*&#F0\"%s!*F0F`qF0F0*&#\"$8\"\"'!)GOF0FdqF0F0*&#\"$J\"FbuF0FiqF0F0*
&#F0\"%?VF0*$FeqF0F0F5*&#F0FbuF0*&)F;FboF0F)F0F0F5*&#F0F[uF0*&FgsF0F,F
0F0F5\"\"*,<*&#F0\"&gX$F0F)F0F5*&#\"$P\"\"'+s!*F0FgoF0F5*&#F0\"&!G<F0F
HF0F5*&#F0\"&S8\"F0FKF0F0*&#\"$T#FivF0FbsF0F0*&#\"#>\"'S9=F0F[pF0F0*&#
F0\"'+%=&F0F^pF0F0*&#F0\"&+C$F0*$FgsF0F0F5*&#\"$R#\"(+)GOF0FfsF0F0*&#
\"$4#\"(+W\"=F0F[tF0F0*&#F0FaxF0*&)F;F\\tF0F)F0F0F5*&#F0\"'+OXF0*&F]vF
0F,F0F0F5#F0\"%S')F0FT-%\"OG6#F0Fer" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 98 "In order to see the parts of the general solution more clearly,
 apply a set of initial conditions." }}{PARA 0 "" 0 "" {TEXT -1 54 "Th
e solution when both initial conditions are zero is:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 25 "ics := y(0)=0, D(y)(0)=0;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%$icsG6$/-%\"yG6#\"\"!F*/--%\"DG6#F(F)F*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "dsolve(\{ode, ics\}, y(x), s
eries);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG+5F'\"\"#F)#
\"\"\"\"\"'\"\"%,$*&#F+\"#5F+-%#lnG6#%\"eGF+!\"\"\"\"&,&#F+\"#gF+*&#F+
\"#IF+*$)F2F)F+F+F6F,,&*&#\"#<\"%g7F+F2F+F6*&#F+\"$E\"F+*$)F2\"\"$F+F+
F6\"\"(,(#F+\"$s'F+*&#FJ\"%?6F+F>F+F6*&#F+FNF+*$)F2F-F+F+F6\"\"),(*&#
\"#T\"&S-$F+F2F+F6*&#F+\"%s!*F+FHF+F+*&#F+\"%?VF+*$)F2F7F+F+F6\"\"*,*#
F+\"%S')F+*&#\"$P\"\"'+s!*F+F>F+F6*&#\"$T#FeoF+FTF+F+*&#F+\"&+C$F+*$)F
2F,F+F+F6F1-%\"OG6#F+\"#6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Rese
t the initial conditions to those of the question::" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 25 "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 33 "dsolve(\{ode, ics\}, y(x)
, series);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG+;F'\"\"\"
\"\"!\"\"%F)#\"\"$\"\"#F.,$*&#F)\"\"'F)-%#lnG6#%\"eGF)!\"\"F-,(#F)\"\"
)F)*&#F)F-F)F3F)F7*&#F)\"#CF)*$)F3F.F)F)F7F+,(*&#\"#6\"$?\"F)F3F)F7*&#
F)\"#5F)F@F)F7*&#F)FFF)*$)F3F-F)F)F7\"\"&,,#F)\"#!)F)*&#F)\"#IF)F3F)F7
*&#\"#<\"$?(F)F@F)F7*&#F)\"#XF)FLF)F7*&#F)FXF)*$)F3F+F)F)F7F2,,*&#\"#f
\"%S]F)F3F)F7*&#F)\"$I'F)F@F)F7*&#\"#>F^oF)FLF)F7*&#F)F^oF)*$)F3FNF)F)
F7*&#F)\"$_#F)FhnF)F7\"\"(,0#F)\"$'*)F)*&#F)\"$O$F)F3F)F7*&#F)FaoF)F@F
)F7*&#F)\"$g$F)FLF)F)*&#F)\"%![%F)FhnF)F7*&#F)\"&?.%F)*$)F3F2F)F)F7*&#
F)\"%!o\"F)FgoF)F7F:,0*&#\"$R\"\"'g47F)F3F)F7*&#F)\"%CIF)F@F)F)*&#\"$J
\"\"'!)GOF)FLF)F)*&#\"#HF^rF)FgoF)F)*&#F]r\"&?2*F)FhnF)F)*&#F)F^rF)*$)
F3F\\pF)F)F7*&#F)\"&gH\"F)F^qF)F7\"\"*,4#F)\"&?:\"F)*&#F)\"%?VF)F3F)F7
*&#Fdo\"'+CIF)F@F)F7*&#Fdo\"&g`%F)FLF)F)*&#\"$r*\"(+)GOF)FhnF)F)*&#\"$
F\"F\\tF)F^qF)F)*&#\"$4#\"'+OXF)FgoF)F)*&#F)F\\tF)*$)F3F:F)F)F7*&#F)\"
'+M6F)FgrF)F7FI-%\"OG6#F)FE" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Ma
ple is unable to obtain an exact solution:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 18 "dsolve(ode, y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#/-%\"yG6#%\"xG-%&DESolG6$<#,*-%%diffG6$-%#_YGF&-%\"$G6$F'\"\"#\"\"
\"*&F'F6-F.6$F0F'F6!\"\"*&)%\"eGF'F6F0F6F6\"\"%F:<#F0" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 24 }{VIEWOPTS 1 
1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }