{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 63 "Computing \+ rational solutions of a linear differential operator." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 90 "L := Dx^2+(11*x^4-7*x^3-4*x-4)/x/(x^3-2)/(x-1)*Dx+2 *(8*x^4-7*x^3+2*x-4)/x/(x^3-2)/(x-1)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG,(*$)%#DxG\"\"#\"\"\"F**&*&,**$)%\"xG\"\"%F*\"#6*&\"\"(F*) F0\"\"$F*!\"\"*&F1F*F0F*F7F1F7F*F(F*F**(F0F*,&*$F5F*F*F)F7F*,&F0F*F*F7 F*F7F**&*&F)F*,*F.\"\")*&F4F*F5F*F7*&F)F*F0F*F*F1F7F*F**(F0F*F:F*)F " 0 "" {MPLTEXT 1 0 37 "sing:=numer(lcoef f(L,Dx)) * denom(L);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%singG*(%\"x G\"\"\",&*$)F&\"\"$F'F'\"\"#!\"\"F'),&F&F'F'F-F,F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "The singularities of L are the roots of this poly nomial, and also the point x=infinity." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "sing:=[seq(i[1],i=factors(sing)[2])];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%singG7%%\"xG,&*$)F&\"\"$\"\"\"F+\"\"#!\"\",&F&F +F+F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 118 "Now every rational solu tion Y of L can be written in the following form, for some integers e[ i] and some polynomial P." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Y:= mul(sing[i]^e[i],i=1..nops(sing)) * P;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"YG**)%\"xG&%\"eG6#\"\"\"F+),&*$)F'\"\"$F+F+\"\"#!\" \"&F)6#F1F+),&F'F+F+F2&F)6#F0F+%\"PGF+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 221 "Notice that the above formula only depends on the variab le \"sing\" which gives the location of the singularities. And sing is the set of factors of lcoeff(L,Dx) and the factors of the denominator s of the coefficients of L." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 344 "To use the above formula, we need to find the \+ possible values for the e[i]. Furthermore we need to know what the deg ree of P could be, we need an upper bound for degree(P). Once we have \+ that, we can substitute unknown coefficients for P. Then Y can be plug ged into the differential equation, we solve for the unknown coefficie nts, and obtain Y." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "The valuation of Y at x=infinity is:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 69 "v_Y_inf := - (degreeP + add(degree(sing[i],x )*e[i],i=1..nops(sing)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(v_Y_in fG,*%(degreePG!\"\"&%\"eG6#\"\"\"F'*&\"\"$F+&F)6#\"\"#F+F'&F)6#F-F'" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 134 "and so this number must be an e xponent of L at x=infinity. Suppose that e[infinity] is the smallest e xponent of L at x=infinity. Then:" }}{PARA 0 "" 0 "" {TEXT -1 24 " v_Y _inf >= e[infinity]" }}{PARA 0 "" 0 "" {TEXT -1 34 "Therefore, degree P can be at most:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "bound_ degreeP := solve(e[infinity] = v_Y_inf, degreeP);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%.bound_degreePG,*&%\"eG6#%)infinityG!\"\"&F'6#\"\" \"F**&\"\"$F-&F'6#\"\"#F-F*&F'6#F/F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Now we need to compute the e[i]." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 804 "smallest_integer_exponent := proc(L, Dx, x, point)\n local T,result;\n # options trace;\n if has(point,x) then\n \+ # point is a polynomial, but what we need is the\n # root of th at polynomial\n RETURN( procname(L, Dx, x, RootOf(point,x) ))\n \+ fi;\n # T is only used as the name of the local parameter.\n # He re we compute all integer exponents (the option\n # restrict_to=\{.. , integer\} throws all non-integer\n # exponents away). Then we take the 'minimal' of\n # these integer exponents.\n result := DEtools [gen_exp](L, [Dx,x], T, x=point,\n 'restrict_to'=\{'minimal', in teger\});\n if result=[] then\n # there are no integer exponen ts, therefore there\n # can not be any rational solutions\n \+ ERROR(\"no nonzero rational solution exists\")\n fi;\n result[1] [1]\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "for i from 1 t o nops(sing) do\n e[i] := smallest_integer_exponent(L, Dx, x, sing[i ])\nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"eG6#\"\"\"\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"eG6#\"\"#!\"#" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>&%\"eG6#\"\"$!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "e[infinity]:=smallest_integer_exponent(L, Dx, x, infi nity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"eG6#%)infinityG\"\"#" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "P:=add(c[i]*x^i,i=0..bound _degreeP);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG,0&%\"cG6#\"\"!\" \"\"*&&F'6#F*F*%\"xGF*F**&&F'6#\"\"#F*)F.F2F*F**&&F'6#\"\"$F*)F.F7F*F* *&&F'6#\"\"%F*)F.F " 0 "" {MPLTEXT 1 0 2 "Y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,0&%\"cG6#\"\"!\"\"\"*&&F&6#F)F)%\"xGF)F)*&&F&6#\"\"# F))F-F1F)F)*&&F&6#\"\"$F))F-F6F)F)*&&F&6#\"\"%F))F-F;F)F)*&&F&6#\"\"&F ))F-F@F)F)*&&F&6#\"\"'F))F-FEF)F)F)*&),&*$F7F)F)F1!\"\"F1F)),&F-F)F)FK F1F)FK" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "with(DEtools): _ Envdiffopdomain:=[Dx,x];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%1_Envdif fopdomainG7$%#DxG%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "e q:=diffop2de(L,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG,(*&*&, **$)%\"xG\"\"%\"\"\"\"\")*&\"\"(F-)F+\"\"$F-!\"\"*&\"\"#F-F+F-F-F,F3F- -%\"yG6#F+F-F-*(F+F-,&*$F1F-F-F5F3F-),&F+F-F-F3F5F-F3F5*&*&,*F)\"#6*&F 0F-F1F-F3*&F,F-F+F-F3F,F3F--%%diffG6$F6F+F-F-*(F+F-F:F-F=F-F3F--FE6$F6 -%\"$G6$F+F5F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "subs(y(x) =Y,eq);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,(*&*&,**$)%\"xG\"\"%\"\"\" \"\")*&\"\"(F+)F)\"\"$F+!\"\"*&\"\"#F+F)F+F+F*F1F+,0&%\"cG6#\"\"!F+*&& F66#F+F+F)F+F+*&&F66#F3F+)F)F3F+F+*&&F66#F0F+F/F+F+*&&F66#F*F+F(F+F+*& &F66#\"\"&F+)F)FIF+F+*&&F66#\"\"'F+)F)FNF+F+F+F+*(F)F+),&*$F/F+F+F3F1F 0F+),&F)F+F+F1F*F+F1F3*&*&,*F'\"#6*&F.F+F/F+F1*&F*F+F)F+F1F*F1F+-%%dif fG6$*&F4F+*&)FRF3F+)FUF3F+F1F)F+F+*(F)F+FRF+FUF+F1F+-Fgn6$Fin-%\"$G6$F )F3F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "normal(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,6*&)%\"xG\"\"(\"\"\"&%\"cG6#\"\"& F*F.*(\"\")F*)F(\"\"'F*&F,6#\"\"%F*F**(\"#OF*&F,6#F2F*)F(F.F*F**(\"\"* F*F:F*&F,6#\"\"$F*F**(\"#?F*F+F*)F(F5F*F**(F0F*FBF*&F,6#\"\"#F*F**(F0F *F3F*)F(F?F*F**(F.F*FHF*&F,6#F*F*F**(F5F*FDF*F(F*!\"\"*&F5F*FJF*FMF**( ),&F(F*F*FMFFF*),&*$FHF*F*FFFMF?F*F(F*FMFM" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 21 "\{coeffs(numer(%),x)\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<),&&%\"cG6#\"\"&!#?*&\"\")\"\"\"&F&6#\"\"#F,!\"\",&&F& 6#\"\"%!\")*&F(F,&F&6#F,F,F0,&&F&6#\"\"'!#O*&\"\"*F,&F&6#\"\"$F,F0,$F7 F4,$F-F4,$F%!\"&,$F2F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "s ols:=subs(solve(%),Y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solsG*&,( &%\"cG6#\"\"!\"\"\"*(\"\"%F+&F(6#\"\"'F+)%\"xG\"\"$F+!\"\"*&F.F+)F2F0F +F+F+*&),&*$F1F+F+\"\"#F4F;F+),&F2F+F+F4F;F+F4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Now verify the answer:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "subs(y(x)=sols,eq);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#,(*&*&,**$)%\"xG\"\"%\"\"\"\"\")*&\"\"(F+)F)\"\"$F+!\"\"*&\"\"#F+F)F +F+F*F1F+,(&%\"cG6#\"\"!F+*(F*F+&F66#\"\"'F+F/F+F1*&F:F+)F)F " 0 "" {MPLTEXT 1 0 10 "normal(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "We see that we find a two dimensional space of rational solutions, because sols has two arbitrary unknowns." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "How can t he exponents be computed? Take for example:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "p:=x^3-2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG ,&*$)%\"xG\"\"$\"\"\"F*\"\"#!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 122 "How can one compute that the smallest integer exponent for this f actor of lcoeff(L,Dx)*denom(L) is -2 (as computed above)?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "p:=RootOf(p,x);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"pG-%'RootOfG6#,&*$)%#_ZG\"\"$\"\"\"F-\"\"#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "t:=x-p;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"tG,&%\"xG\"\"\"-%'RootOfG6#,&*$)%#_ZG\"\"$F'F'\" \"#!\"\"F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "subs(y(x)=t^n ,eq);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&*&,**$)%\"xG\"\"%\"\"\"\" \")*&\"\"(F+)F)\"\"$F+!\"\"*&\"\"#F+F)F+F+F*F1F+),&F)F+-%'RootOfG6#,&* $)%#_ZGF0F+F+F3F1F1%\"nGF+F+*(F)F+,&*$F/F+F+F3F1F+),&F)F+F+F1F3F+F1F3* &*&,*F'\"#6*&F.F+F/F+F1*&F*F+F)F+F1F*F1F+-%%diffG6$F4F)F+F+*(F)F+F?F+F BF+F1F+-FJ6$F4-%\"$G6$F)F3F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "normal(%/t^n);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*&,fn*(%\"nG\" \"\")%\"xG\"\"&F'-%'RootOfG6#,&*$)%#_ZG\"\"$F'F'\"\"#!\"\"F'!#6**\"\"% F'F&F')F)F3F'F+F'F'*(F3F')F&F3F'F)F'F4*&\"\")F')F+F3F'F4*(F7F'F&F'F8F' F4*(F3F'F&F')F)F2F'F4*(\"\"'F'F&F'F)F'F'*&F:F')F)FBF'F'*(F3F'F:F'F(F'F 4*&F:F')F)F7F'F'*(F3F'F:F'F@F'F4*(F7F'F:F'F8F'F'*(FBF'F&F'FGF'F'*(\"#5 F'F&F'FDF'F'*(\"#;F'F&F'F(F'F4*&F " 0 "" {MPLTEXT 1 0 21 "evala(subs(x=T+p,%));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*&,jo*(%\"nG\"\"\"-%'RootOfG6#,&*$)%#_ZG\"\"$F'F'\"\"#!\"\"F')% \"TGF/F'!#i**\"#F'F1* *\"#))F'F&F'F>F'F7F'F1**\"\"'F')F&F0F'F3F'F>F'F'**\"#CF'FHF'F3F'F(F'F' **\"#:F'FHF'F2F'F>F'F'**FGF'FHF'F:F'F(F'F'**\"#yF'F&F'F(F'F3F'F'*&FGF' FHF'F'**\"#5F'FHF'F2F'F(F'F1*(FJF'F&F'F(F'F1*&\"$K\"F'F7F'F'*(\"#OF'F3 F'F(F'F'*(\"$3\"F'F&F'F3F'F1*(\"#7F'F&F'F>F'F'*(\"$y\"F'F&F'F7F'F'*(\" #'*F'F2F'F>F'F'*(FGF'F&F'F2F'F'*&\"#;F')F3\"\"&F'F'*(\"#kF'F:F'F(F'F'* (FWF'FHF'F3F'F1*(FenF'FHF'F(F'F1*&FHF'F2F'F'*&FHF'F]oF'F'*(\"#QF'FHF'F 7F'F'*(FGF'FHF'F>F'F'*(F0F'FHF'F:F'F1**\"#?F'FHF'F7F'F>F'F1*(F\\oF'F&F 'F:F'F1*(FRF'F&F'F]oF'F'**F;F'FHF'F7F'F(F'F'*&FWF'F3F'F1*&FenF'F&F'F'* *FLF'F&F'F>F'F3F'F'F'**,&F3F'F(F'F',(*$F7F'F'*(F/F'F3F'F(F'F'*&F/F'F>F 'F'F'),(F3F'F(F'F'F1F0F'F7F'F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "The denominator has a factor T^2 and the numerator does not. So t he series expansion of this at T=0 is:" }}{PARA 0 "" 0 "" {TEXT -1 50 " A_2*T^(-2) + A_1*T^(-1) + A0*T^0 + A1*T^1 + ...." }}{PARA 0 "" 0 " " {TEXT -1 136 "and the tcoeff of this, which is A_2, is the indicial \+ equation of L at x=p. We can find it by multiplying by T^2, then subst ituting T=0." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "indicial_eq uation := evala(subs(T=0,evala(% * T^2)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"nG\"\"\",&F$F%\"\"#F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "exponents := \{solve(%,n)\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$!\"#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "integer_exponents := select(type,%,integer);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$!\"#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "smallest := min(op(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"#" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Likewise, if we want to compute \+ the exponents at:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "p:=inf inity;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG%)infinityG" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "then the local parameter is:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "t:=1/x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"tG*&\"\"\"F&%\"xG!\"\"" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 21 "subs(y(x) = t^n, eq);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&*&,**$)%\"xG\"\"%\"\"\"\"\")*&\"\"(F+)F)\"\"$F+!\"\"*&\"\"#F +F)F+F+F*F1F+)*&F+F+F)F1%\"nGF+F+*(F)F+,&*$F/F+F+F3F1F+),&F)F+F+F1F3F+ F1F3*&*&,*F'\"#6*&F.F+F/F+F1*&F*F+F)F+F1F*F1F+-%%diffG6$F4F)F+F+*(F)F+ F8F+F;F+F1F+-FC6$F4-%\"$G6$F)F3F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "normal(%/t^n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&, B*$)%\"xG\"\"&\"\"\"\"#;*&\"#9F))F'\"\"%F)!\"\"*&F.F))F'\"\"#F)F)*&\" \")F)F'F)F/*(\"#5F)%\"nGF)F&F)F/*(F*F)F7F)F-F)F)*(\"\"'F)F7F))F'\"\"$F )F/*(F2F)F7F)F1F)F)*&F:F)F7F)F/*&)F7F2F)F&F)F)*(F2F)F@F)F-F)F/*&F@F)F; F)F)*(F2F)F@F)F1F)F/*(F.F)F@F)F'F)F)*&F2F)F@F)F/*(F.F)F7F)F'F)F)F)*(F1 F),&*$F;F)F)F2F/F)),&F'F)F)F/F2F)F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 134 "Now t=1/x, so x=1/T where T is used as the name of the local p arameter (I can't use the name t because t is already assigned a value )." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "AA:=normal(subs(x=1/T ,%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#AAG*&*&,B!#;\"\"\"*&\"#9F) %\"TGF)F)*&\"\"%F))F,\"\"$F)!\"\"*&\"\")F))F,F.F)F)*&\"#5F)%\"nGF)F)*( \"#;F)F7F)F,F)F1*(\"\"'F)F7F))F,\"\"#F)F)*(F=F)F7F)F/F)F1*(F;F)F7F))F, \"\"&F)F)*$)F7F=F)F1*(F=F)FCF)F,F)F)*&FCF)F " 0 "" {MPLTEXT 1 0 46 "indicial_equation := subs(T=0,norma l(AA/T^2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%2indicial_equationG,( \"#;\"\"\"*&\"#5F'%\"nGF'!\"\"*$)F*\"\"#F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "series(AA,T=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+-%\"TG,(\"#;\"\"\"*&\"#5F'%\"nGF'!\"\"*$)F*\"\"#F'F'F.,&\"#=F'*&\" \"%F'F*F'F+\"\"$,&\"#?F'*&F2F'F*F'F+F2,&\"#eF'*&\"#AF'F*F'F+\"\"&-%\"O G6#F'\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "exponents:=so lve(indicial_equation,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*expone ntsG6$\"\")\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "The smallest \+ one is 2, like the procedure given above had already determined." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "e[infinity];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "55 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }