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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "# Series solution of odes
\n\n# filename = 04.7.mws\n\nExample 1" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 
"\nsola:=dsolve((1-x^2)*diff(y(x),x$2)-2*x*diff(y(x),x)+k*(k+1)*y(x)=0
,y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "solb:=dsolve((1-
x^2)*diff(y(x),x$2)-2*x*diff(y(x),x)+k*(k+1)*y(x)=0,y(x),'type=series'
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "solve(2*a[2]+k*(k+1)*
a[0]=0,a[2]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "solve(-2*a
[1]+k*(k+1)*a[1]+6*a[3]=0,a[3]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 72 "genform:=solve((n+1)*(n+1)*a[n+2]+(-n*(n-1)-2*n+k*(k+
1))*a[n]=0,a[n+2]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "subs
(n=n-2,genform);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "a[n]:=-
(-(n-2)^2-n+2+k^2+k)*a[n-2]/((n-1)^2):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "seq(a[n],n=1..5);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 84 "a:= proc(n) option remember;\n            -(-(n-2)^2-
n+2+k^2+k)*a(n-2)/((n-1)^2) end:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 55 "a(1):=a[1]:\na(0):=a[0]:\narray([seq([i,a(i)],i=2..10
)]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(orthopoly);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "?P" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 27 "toplot:=seq(P(n,x),n=0..5);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 31 "plot(\{toplot\},x=-2..2,y=-2..2);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "array([seq([seq(int(P(n,x)*P
(m,x),x=-1..1),n=0..6)],m=0..6)]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {SPREADSHEET {NAME "" } 
{ROWHEIGHTS 1 61 2 55 3 55 4 65 5 95 } {COLWIDTHS 2 288 3 290 } 
{SSOPTS {CELLOPTS 2 10 4 2 1 255 255 255 }1 }700 384 384 {CELL 1 1 
{CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "n" 20 "6#%\"nG" }0 }{CELL 1 
2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "(1-x^2)*diff(y(x),`$`(x,2))
-2*x*diff(y(x),x)+~A1*(~A1+1)*y(x) = 0" 20 "6#/,(*&,&\"\"\"F'*$)%\"xG
\"\"#F'!\"\"F'-%%diffG6$-%\"yG6#F*-%\"$G6$F*F+F'F'*&F*F'-F.6$F0F*F'!\"
#*(%\"nGF',&F;F'F'F'F'F0F'F'\"\"!" }0 }{CELL 1 3 {CELLOPTS 0 -1 -1 0 
0 0 0 0 }{R5MATHOBJ "dsolve(~B1,y(x))" 20 "6#/-%\"yG6#%\"xG,&*&%$_C1G
\"\"\"-%*LegendrePG6$%\"nGF'F+F+*&%$_C2GF+-%*LegendreQGF.F+F+" }0 }
{CELL 2 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "1" 20 "6#\"\"\"" }
0 }{CELL 2 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "(1-x^2)*diff(y(x
),`$`(x,2))-2*x*diff(y(x),x)+~A2*(~A2+1)*y(x) = 0" 20 "6#/,(*&,&\"\"\"
F'*$)%\"xG\"\"#F'!\"\"F'-%%diffG6$-%\"yG6#F*-%\"$G6$F*F+F'F'*&F*F'-F.6
$F0F*F'!\"#F0F+\"\"!" }0 }{CELL 2 3 {CELLOPTS 0 -1 -1 0 0 0 0 0 }
{R5MATHOBJ "dsolve(~B2,y(x))" 20 "6#/-%\"yG6#%\"xG,&*&%$_C1G\"\"\"F'F+
F+*&%$_C2GF+,&*&F'F+-%#lnG6#,$*&,&F'F+F+F+F+,&!\"\"F+F'F+F7F7F+#F+\"\"
#F7F+F+F+" }0 }{CELL 3 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "2" 
20 "6#\"\"#" }0 }{CELL 3 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "(1
-x^2)*diff(y(x),`$`(x,2))-2*x*diff(y(x),x)+~A3*(~A3+1)*y(x) = 0" 20 "6
#/,(*&,&\"\"\"F'*$)%\"xG\"\"#F'!\"\"F'-%%diffG6$-%\"yG6#F*-%\"$G6$F*F+
F'F'*&F*F'-F.6$F0F*F'!\"#F0\"\"'\"\"!" }0 }{CELL 3 3 {CELLOPTS 0 -1 
-1 0 0 0 0 0 }{R5MATHOBJ "dsolve(~B3,y(x))" 20 "6#/-%\"yG6#%\"xG,&*&%$
_C1G\"\"\",&*$)F'\"\"#F+\"\"$!\"\"F+F+F+*&%$_C2GF+,(*&-%#lnG6#,$*&,&F'
F+F+F+F+,&F1F+F'F+F1F1F+F.F+#F0\"\"%F6#F1F>F'#!\"$F/F+F+" }0 }{CELL 4 
1 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "3" 20 "6#\"\"$" }0 }{CELL 
4 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "(1-x^2)*diff(y(x),`$`(x,2
))-2*x*diff(y(x),x)+~A4*(~A4+1)*y(x) = 0" 20 "6#/,(*&,&\"\"\"F'*$)%\"x
G\"\"#F'!\"\"F'-%%diffG6$-%\"yG6#F*-%\"$G6$F*F+F'F'*&F*F'-F.6$F0F*F'!
\"#F0\"#7\"\"!" }0 }{CELL 4 3 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ 
"dsolve(~B4,y(x))" 20 "6#/-%\"yG6#%\"xG,&*&%$_C1G\"\"\",&*$)F'\"\"$F+
\"\"&F'!\"$F+F+*&%$_C2GF+,**&-%#lnG6#,$*&,&F'F+F+F+F+,&!\"\"F+F'F+F=F=
F+F.F+#F0\"\"%*&F'F+F6F+#F1F?*$)F'\"\"#F+#!\"&FD#FDF/F+F+F+" }0 }
{CELL 5 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "4" 20 "6#\"\"%" }0 
}{CELL 5 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "(1-x^2)*diff(y(x),
`$`(x,2))-2*x*diff(y(x),x)+~A5*(~A5+1)*y(x) = 0" 20 "6#/,(*&,&\"\"\"F'
*$)%\"xG\"\"#F'!\"\"F'-%%diffG6$-%\"yG6#F*-%\"$G6$F*F+F'F'*&F*F'-F.6$F
0F*F'!\"#F0\"#?\"\"!" }0 }{CELL 5 3 {CELLOPTS 0 -1 -1 0 0 0 0 0 }
{R5MATHOBJ "dsolve(~B5,y(x))" 20 "6#/-%\"yG6#%\"xG,&*&%$_C1G\"\"\",(*$
)F'\"\"%F+\"#N*$)F'\"\"#F+!#I\"\"$F+F+F+*&%$_C2GF+,,*&-%#lnG6#,$*&,&F'
F+F+F+F+,&!\"\"F+F'F+FAFAF+F.F+#F0\"#;*&F:F+F2F+#!#:\"\")F:#F5FC*$)F'F
5F+#!#NFGF'#\"#b\"#CF+F+" }0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Ex
ample 2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "eq:=x^2*diff(y(x
),x$2)+x*diff(y(x),x)+(x^2-mu^2)*y(x)=0:\nwith(DEtools):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "indicialeq(eq,x,0,y(x));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "a:='a':\nsolve(((n+mu)*(n+mu
-1)+(n+mu)-mu^2)*a[n]+a[n-2]=0,a[n]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 78 "a:= proc(n) option remember;\n            -a(n-2)/(n*
(n+2*mu)) end:\na(0):=a[0]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
55 "nvals:=seq(2*i,i=1..5):\narray([seq([n,a(n)],n=nvals)]);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dsolve(eq,y(x));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "dsolve(eq,y(x),'type=series');" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "plot(\{seq(BesselJ(mu,x),mu=
0..4)\},x=0..10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "plot(
\{seq(BesselY(mu,x),mu=0..4)\},x=0..10,y=-9..1);" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "30" 0 }{VIEWOPTS 1 1 0 1 1 1803 
1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }