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" {TEXT 256 51 "Power series solutions with polynomial coefficients" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "L := (3*x^2-6*x+3)*diff(dif
f(y(x),x),x)+(12*x-12)*diff(y(x),x)+6*y(x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"LG,(*&,(*$%\"xG\"\"#\"\"$F)!\"'F+\"\"\"F--%%diffG6$
-F/6$-%\"yG6#F)F)F)F-F-*&,&F)\"#7!#7F-F-F1F-F-F3\"\"'" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "`Slode/polycoeffs`(L,y(x));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#<#-%$SumG6$*&,(*&&%#_CG6#\"\"\"F-%\"nG
F-F-F*!\"\"&%#_BGF,F-F-)%\"xGF.F-/F.;\"\"!%)infinityG" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 
257 49 "Power series solutions with rational coefficients" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "L1 := 2*x*(x-1)*diff(y(x),x$2) + (7
*x-3)*diff(y(x),x) + 2*y(x) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#L1
G,(*(%\"xG\"\"\",&F'F(!\"\"F(F(-%%diffG6$-F,6$-%\"yG6#F'F'F'F(\"\"#*&,
&F'\"\"(!\"$F(F(F.F(F(F0F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
27 "`Slode/ratcoeffs`(L1,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#*
&&%#_CG6#\"\"!\"\"\"-%$SumG6$*(,&%\"nGF)F)F)F),&F/\"\"#F)F)!\"\")%\"xG
F/F)/F/;F(%)infinityGF)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 258 55 "Power series solutions wit
h hypergeometric coefficients" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 64 "L1 := 2*x*(x-1)*diff(y(x),x$2) + (7*x-3)*diff(y(x),x) + 2*y(x) ;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#L1G,(*(%\"xG\"\"\",&F'F(!\"\"F(
F(-%%diffG6$-F,6$-%\"yG6#F'F'F'F(\"\"#*&,&F'\"\"(!\"$F(F(F.F(F(F0F3" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "`Slode/hypercoeffs`(L1,y(x
));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<%,$*(&%#_BG6#\"\"\"F)%#PiG#!\"
\"\"\"#-%$SumG6$*,)#F)F-%\"nGF))F-F4F)-%&GAMMAG6#,&F4F)F3F)F)-F76#,&F4
F)F)F)F,),&%\"xGF)F)F)F4F)/F4;\"\"!%)infinityGF)F-,$*&&%#_CGF(F)-F/6$*
,-F76#,&F4F)F-F)F)-F76#,&F4F)F3F)F)-F76#,&F4F)#\"\"$F-F)F,-F76#,&F4F)F
)F)F,)F?F4F)/F4FAF)F3*(&FGF(F)F*F+-F/6$*,)F+F4F))F-F4F)-F76#,&F4F)F3F)
F)-F76#,&F4F)F)F)F,),&F?F)F,F)F4F)/F4FAF)" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 259 66 "Power se
ries solutions with eventually hypergeometric coefficients" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "L2 := (3-x)*diff(y(x),x$2)-diff(y(x
),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#L2G,&*&,&\"\"$\"\"\"%\"xG!
\"\"F)-%%diffG6$-F-6$-%\"yG6#F*F*F*F)F)F/F+" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 29 "`Slode/hypercoeffs`(L2,y(x));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#<$&%#_BG6#\"\"!,&F$\"\"\"*&&%#_CG6#F)F)-%$SumG6$*(-%&GA
MMAG6#%\"nGF)-F36#,&F5F)F)F)!\"\"),&%\"xGF)!\"#F)F5F)/F5;F)%)infinityG
F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "`Slode/hyper_at_poi
nt`(L2,y(x),_A,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&&%#_AG6#\"\"!
\"\"\"*&&%#_CG6#F(F(-%$SumG6$**)#F(\"\"$%\"nGF(-%&GAMMAG6#F4F(-F66#,&F
4F(F(F(!\"\")%\"xGF4F(/F4;F(%)infinityGF(F3" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 38 "`Slode/hyper_at_point`(L2,y(x),_A,10);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,&&%#_AG6#\"\"!\"\"\"*&&%#_CG6#F(F(-%$SumG6$
**)#!\"\"\"\"(%\"nGF(-%&GAMMAG6#F5F(-F76#,&F5F(F(F(F3),&%\"xGF(!#5F(F5
F(/F5;F(%)infinityGF(!\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 260 60 "Power series solutions wi
th eventually rational coefficients" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "L2 := (3-x)*diff(y(x),x$2)-diff(y(x),x);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%#L2G,&*&,&\"\"$\"\"\"%\"xG!\"\"F)-%%diffG6$-F
-6$-%\"yG6#F*F*F*F)F)F/F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
27 "`Slode/ratcoeffs`(L2,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#,
&&%#_BG6#\"\"!\"\"\"*&&%#_CGF'F)-%$SumG6$*&%\"nG!\"\"),&%\"xGF)!\"#F)F
1F)/F1;F)%)infinityGF)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 261 52 "m-Sparse power series solu
tions at the given m-point" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
30 "L3:=diff(y(x),x,x)+(x-1)*y(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%#L3G,&-%%diffG6$-F'6$-%\"yG6#%\"xGF.F.\"\"\"*&,&!\"\"F/F.F/F/F+F/F/
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "`Slode/msparse_at_point
`(L3,y(x),v(n),3,1,C);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<$7$,(&%\"CG
6#\"\"!\"\"\"*&F&F*,&!\"\"F*%\"xGF*\"\"$#F-\"\"'-%$SumG6$*&-%\"vG6#,$%
\"nGF/F*)F,F9F*/F:;\"\"#%)infinityGF*,&-F76#,&F:F/!\"$F*F**(F6F*,&F:F/
F-F*F*F:F*F/7$,(*&&F'6#F*F*F,F*F**&FJF*F,\"\"%#F-\"#7-F36$*&-F76#,&F:F
/F*F*F*)F,FUF*F<F*,&-F76#,&F:F/!\"#F*F**(FSF*F:F*FUF*F/" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 
262 31 "m-Sparse power series solutions" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 147 "L4:=(x-1)*y(x)+ diff(y(x),x$2)+ (-x+1)*diff(y(x),x$3
)+ (-x+1)*diff(y(x),x$4)- diff(y(x),x$5)- diff(y(x),x$6)+ (x-1)*diff(y
(x),x$7)+ diff(y(x),x$9);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#L4G,2*
&,&!\"\"\"\"\"%\"xGF)F)-%\"yG6#F*F)F)-%%diffG6$-F/6$F+F*F*F)*&,&F*F(F)
F)F)-F/6$F.F*F)F)*&F4F)-F/6$F5F*F)F)-F/6$F8F*F(-F/6$F:F*F(*&F'F)-F/6$F
<F*F)F)-F/6$-F/6$F?F*F*F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
38 "res:=`Slode/msparse`(L4,y(x),v(n),_C):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 10 "nops(res);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"*
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "res[1];" }}{PARA 12 "" 
1 "" {XPPMATH 20 "6#7$,:&%#_CG6#\"\"!\"\"\"*&&F&6#F)F),&!\"\"F)%\"xGF)
F)F)*&&F&6#\"\"#F)F-F3F)*&&F&6#\"\"$F)F-F7F)*&,*F+#F)\"#7&F&6#\"\"&!#5
&F&6#\"\"(!$5#&F&6#\"\")\"%!o\"F)F-\"\"%F)*&F<F)F-F>F)*&,&F1#F)\"$g$FD
!#cF)F-\"\"'F)*&F@F)F-FBF)*&FDF)F-FFF)*&,&F<#F)\"%CIFD#F.\"\"*F)F-FWF)
*&,,F%#F.\"(+)GOF+#F)\"(+W\"=F<#F.\"&?^\"F@#F.\"$?(F1FfnF)F-\"#5F)-%$S
umG6$*&-%\"vG6#,&%\"nGF7F)F)F))F-FdoF)/Feo;FH%)infinityGF),&-Fbo6#,&Fe
oF7!\"#F)F.**FaoF),&FeoF7F.F)F)FeoF)FdoF)F7" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 7 "res[2];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7$,,&%#_
CG6#\"\"!\"\"\"*&&F&6#\"\"#F)%\"xGF-F)*&F%F)F.\"\"%#F)\"#C*&F+F)F.\"\"
'#F)\"$g$-%$SumG6$*&-%\"vG6#,$%\"nGF-F))F.F>F)/F?;F0%)infinityGF),&-F<
6#,&F?F-!\"%F)!\"\"*,F;F),&F?F-!\"$F)F),&F?F-!\"#F)F),&F?F-FIF)F)F?F)F
-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "res[3];" }}{PARA 12 "" 
1 "" {XPPMATH 20 "6#7$,:&%#_CG6#\"\"!\"\"\"*&&F&6#F)F),&!\"\"F)%\"xGF)
F)F)*&&F&6#\"\"#F)F-F3F)*&&F&6#\"\"$F)F-F7F)*&,(F%#F)\"#CF1#F.\"#7&F&6
#\"\")\"%!o\"F)F-\"\"%F)*&&F&6#\"\"&F)F-FFF)*&,&F1#F)\"$g$F>!#cF)F-\"
\"'F)*&&F&6#\"\"(F)F-FPF)*&F>F)F-F@F)*&,&FD#F)\"%CIF>#F.\"\"*F)F-FWF)*
&,,F+#F.\")+%e*>F%#F)\")+o\"*RF1FZFD#F)\"'?j;FN#F)\"%?zF)F-\"#6F)-%$Su
mG6$*&-%\"vG6#,&%\"nGF7F3F)F))F-FdoF)/Feo;FB%)infinityGF),&-Fbo6#,&Feo
F7F.F)F.**FaoF)FeoF),&FeoF7F)F)F)FdoF)F7" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 264 49 "m-Sparse m
-hypergeometric power series solutions " }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "L3:=diff(y(x),x,x)+(x-1)*y(x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#L3G,&-%%diffG6$-F'6$-%\"yG6#%\"xGF.F.\"\"\"*&,&F.F/!
\"\"F/F/F+F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "`Slode/mh
ypercoeffs`(L3,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$,&&%#_BG6#
\"\"!\"\"\"*(F%F)-%&GAMMAG6##\"\"#\"\"$F)-%$SumG6$*,)#!\"\"F0%\"nGF)-F
,6#,&F8F)F)F)F7)F0F8F7-F,6#,&F8F)F.F)F7),&%\"xGF)F7F),$F8F0F)/F8;F)%)i
nfinityGF)F),&*&&%#_CG6#F)F)FAF)F)*,FIF)%#PiGF)F0#F)F/F+F7-F26$*,F5F)F
<F7-F,6#,&F8F)#\"\"%F0F)F7F9F7)FA,&F8F0F)F)F)FDF)#F/\"\"*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "L5 := (2+x^2)*diff(y(x),x$3)- 2*dif
f(y(x),x$2)*x+ (2+x^2)*diff(y(x),x)- 2*x*y(x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#L5G,**&,&\"\"#\"\"\"*$%\"xGF(F)F)-%%diffG6$-F-6$-F-6
$-%\"yG6#F+F+F+F+F)F)*&F/F)F+F)!\"#*&F'F)F1F)F)*&F+F)F3F)F7" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "`Slode/mhypercoeffs`(L3,y(x)
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$,(*&&%#_CG6#\"\"\"F)%\"xGF)F)*
&&%#_BG6#\"\"#F)F*F/F)*(F&F)%#PiG#F)F/-%$SumG6$**)#!\"\"\"\"%%\"nGF)-%
&GAMMAG6#,&F;F)#\"\"$F/F)F9-F=6#,&F;F)F)F)F9)F*,&F;F/F)F)F)/F;;F)%)inf
inityGF)F2,(&%#_CGF(F)F+F)*(FKF)F1F2-F46$**F7F)-F=6#,&F;F)F2F)F9FBF9)F
*,$F;F/F)/F;;F/FIF)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "`S
lode/mhyper_at_point`(L3,y(x),2,-1);" }}{PARA 12 "" 1 "" {XPPMATH 20 "
6#<$,,&%#_BG6#\"\"\"#!\"$\"\"#&F&6#F+!\"#*&F%F(,&%\"xGF(F(F(F(F(*&F,F(
F0F+F(*(,&F%F(F,F+F(%#PiG#F(F+-%$SumG6$*,)#!\"\"F+%\"nGF(-%&GAMMAG6#,&
F>F(F(F(F=)F+F>F=-F@6#,&F>F(F6F(F=)F0,$F>F+F(/F>;F+%)infinityGF(F=,,F,
F(*&,&&%#_CGF'F(F,F.F(F0F(F(F2F(*&FOF(F0\"\"$#F=\"\"'*(FOF(F5F6-F86$*,
F;F(FCF=-F@6#,&F>F(#FRF+F(F=F?F=)F0,&F>F+F(F(F(FIF(F6" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "`Slode/mhyper_at_point`(L3,y(x),2,-
1/2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<$,,&%#_BG6#\"\"\"#!\"*\"\"%&
F&6#\"\"#!\"#*&F%F(,&%\"xGF(#F(F.F(F(F(*&F,F(F1F.F(*(,&F%F(F,F(F(%#PiG
F3-%$SumG6$*,)#!\"\"F.%\"nGF(-%&GAMMAG6#,&F?F(F(F(F>)F.F?F>-FA6#,&F?F(
F3F(F>)F1,$F?F.F(/F?;F.%)infinityGF(F/,,F,#F(F+*&,&&%#_CGF'F(F,F>F(F1F
(F(F4F(*&FQF(F1\"\"$#F>\"\"'*(FQF(F7F3-F96$*,F<F(FDF>-FA6#,&F?F(#FTF.F
(F>F@F>)F1,&F?F.F(F(F(FJF(F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
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