{VERSION 5 0 "SUN SPARC SOLARIS" "5.0" } {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 Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 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 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 48 "Exponents \+ of a differential operator at a point." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 81 "Take the point x=0. The functions we \+ consider now are formal power series at x=0:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "y = sum(a[n]*x ^n,i = N .. infinity)" "6#/%\"yG-%$sumG6$*&&%\"aG6#%\"nG\"\"\")%\"xGF, F-/%\"iG;%\"NG%)infinityG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 25 "where N is an element of " }{TEXT 257 1 "Z" } {TEXT -1 248 "=\{...,-3,-2,-1,0,1,2,3,....\}, and where a[n] are compl ex numbers. Right now we do not worry about convergence (formal series means: no convergence conditions on the a[i]). We will now apply a li near differential operator L on that and we get\nL(y) =" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "L(sum(a[n]*x^n,n = N .. infinity)) = sum(a[n]*L(x^n),x = N .. infinity);" "6#/-%\"LG6#-% $sumG6$*&&%\"aG6#%\"nG\"\"\")%\"xGF.F//F.;%\"NG%)infinityG-F(6$*&&F,6# F.F/-F%6#)F1F.F//F1;F4F5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 42 "These latter two are equal because L is a " } {TEXT 258 1 "C" }{TEXT -1 59 "-linear map. Now lets focus on one parti cular term: L(x^n)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "appl y_L := proc(L, y, Dx,x)\n local i;\n add(coeff(L,Dx,i) * diff(y,[x$i ]), i=0..degree(L,Dx))\nend;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(app ly_LGf*6&%\"LG%\"yG%#DxG%\"xG6#%\"iG6\"F--%$addG6$*&-%&coeffG6%9$9&8$ \"\"\"-%%diffG6$9%7#-%\"$G6$9'F7F8/F7;\"\"!-%'degreeG6$F5F6F-F-F-" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "L:=x*Dx^4+6*Dx^3+x/(x-1)*Dx^ 2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG,(*&%\"xG\"\"\")%#DxG\"\"% F(F(*&\"\"'F()F*\"\"$F(F(*(F'F(,&F'F(F(!\"\"F2F*\"\"#F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "apply_L( L, x^n, Dx,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*(%\"xG\"\"\",&F%F&F&!\"\"F(,&*()F%%\"nGF& F,\"\"#F%!\"#F&*(F+F&F,F&F%F.F(F&F&**\"\"'F&F+F&F,\"\"$F%!\"$F&**\"#=F &F+F&F,F-F%F3F(**\"#7F&F+F&F,F&F%F3F&*&F%F&,**(F+F&F,\"\"%F%!\"%F&**F1 F&F+F&F,F2F%F " 0 "" {MPLTEXT 1 0 14 "normal(%/x^n);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#**%\"xG!\"$%\"nG\"\"\",2*&)F$\"\"#F'F&F'F'*$F*F' !\"\"*(\"\"(F'F&F'F$F'F-*&F/F'F&F'F'*&\"\"'F'F$F'F'F2F-*&)F&\"\"$F'F$F 'F'*$F4F'F-F',&F$F'F'F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "convert(series(%,x=0),polynom);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,2*(%\"nG\"\"\",(\"\"'!\"\"*&\"\"(F&F%F&F&*$)F%\"\"$F&F)F&%\"xG!\"$F) *(F%F&,&F%F&F&F)F&F/F)F)*&F%F&F2F&F)*(F%F&F2F&F/F&F)*(F%F&F2F&)F/\"\"# F&F)*(F%F&F2F&)F/F.F&F)*(F%F&F2F&)F/\"\"%F&F)*(F%F&F2F&)F/\"\"&F&F)" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "lowest_deg, lowest_coeff : = ldegree(%,x), tcoeff(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%+l owest_degG%-lowest_coeffG6$!\"$,$*&%\"nG\"\"\",(\"\"'!\"\"*&\"\"(F,F+F ,F,*$)F+\"\"$F,F/F,F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "fa ctor(lowest_coeff);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**%\"nG\"\"\",& F$F%F%!\"\"F%,&F$F%\"\"#F'F%,&F$F%\"\"$F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "solve(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"! \"\"\"\"\"#!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 212 "We see that t he lowest power of x of L(x^n)/x^n written as a power series in x equa ls -3, unless n in \{0,1,2, -3\} because then the lowest_coeff=0 in wh ich case the lowest_deg is not -3 but will be higher than -3." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 109 "In other words: The valuation at x=0 (i.e. lowest power of x-0 with a non-zero coefficient) of L(x^n)/x^n is:" }}{PARA 0 "" 0 "" {TEXT -1 29 " > \+ -3 if n in \{0,1,2, -3\}" }}{PARA 0 "" 0 "" {TEXT -1 19 " = -3 oth erwise." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Hence the valuation of L(x^n) at x=0 is:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "v0( L(x^n) ) > n-3 if n in \+ \{0,1,2, -3\}" }}{PARA 0 "" 0 "" {TEXT -1 44 "v0( L(x^n) ) = n-3 if \+ n not in \{0,1,2,-3\}." }}{PARA 0 "" 0 "" {TEXT -1 31 "Now consider ag ain L(y) where: " }{XPPEDIT 18 0 "y = sum(a[n]*x^n,i = N .. infinity) " "6#/%\"yG-%$sumG6$*&&%\"aG6#%\"nG\"\"\")%\"xGF,F-/%\"iG;%\"NG%)infin ityG" }}{PARA 0 "" 0 "" {TEXT -1 51 "Assume that a[N]<>0, in other wor ds: v0(y)=N. Then:" }}{PARA 0 "" 0 "" {TEXT -1 39 "v0( L(a[n] * x^n) ) >= n-3 for all n" }}{PARA 0 "" 0 "" {TEXT -1 119 "Note that if a[n ]=0 then this number is infinity, which is also >= n-3. We see now tha t we have only two possibilities:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 56 "Case 1). v0(L)=N is in \{0,1,2, -3\}. Th en v0(L(y)) > N-3" }}{PARA 0 "" 0 "" {TEXT -1 58 "Case 2). v0(L)=N is \+ not in \{0,1,2,-3\}. Then v0(L(y)) = N-3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 212 "In case 2, we see that v0(L(y)) i s N-3, which is less than infinity. If the valuation is not infinity, \+ then the function is not 0. In other words: L(y)<>0 in case 2. Therefo re, y is not a solution of L in case 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "So: only in case 1 can y be a soluti on of L. The numbers \{0,1,2,-3\} are called the exponents of L." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 60 "Computin g the indicial equation of an operator L at a point." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "L := Dx^2-4*(3*x^4-2*x^3+8*x^2-9*x+3)/(x- 1)^2/(x^2+1)/x^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG,&*$)%#DxG \"\"#\"\"\"F**,\"\"%F*,,*&\"\"$F*)%\"xGF,F*F**&F)F*)F1F/F*!\"\"*&\"\") F*)F1F)F*F**&\"\"*F*F1F*F4F/F*F*,&F1F*F*F4!\"#,&*$F7F*F*F*F*F4F1F;F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "points:=[-1, 0, 1, infini ty];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'pointsG7&!\"\"\"\"!\"\"\"%) infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "for p in point s do t[p]:=`if`(p=infinity,1/x,x-p) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"tG6#!\"\",&%\"xG\"\"\"F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"tG6#\"\"!%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"tG6#\" \"\",&%\"xGF'F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"tG6#%)inf inityG*&\"\"\"F)%\"xG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "t[p ] is the local parameter at x=p" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "for p in points do K[p]:=series(normal(apply_L(L, t[p]^n, Dx, x)/t[p]^n),x=p) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"KG6#!\"\" +/,&%\"xG\"\"\"F+F+,&%\"nGF'*$)F-\"\"#F+F+!\"##!#DF0\"\"!#!#dF0F+#!$l$ \"\")F0#!$X#\"\"%\"\"$-%\"OG6#F+F<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >&%\"KG6#\"\"!+1%\"xG,(\"#7!\"\"*$)%\"nG\"\"#\"\"\"F1F/F,!\"#F+F,\"#;F '!\")F1!#CF0!#7\"\"$-%\"OG6#F1\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>&%\"KG6#\"\"\"+1,&%\"xGF'F'!\"\",(\"\"'F+%\"nGF+*$)F.\"\"#F'F'!\"#! \")F+\"\"&\"\"!F4F'#!#RF1F1\"#N\"\"$-%\"OGF&\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"KG6#%)infinityG,,*&,(\"#7!\"\"*$)%\"nG\"\"#\"\"\"F 1F/F1F1%\"xG!\"#F1*&\"#;F1F2!\"$F,*&\"#SF1F2!\"%F,*&\"#OF1F2!\"&F,-%\" OG6#*&F1F1*$)F2\"\"'F1F,F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "Now write them in terms of the local parameter t[p] using the variable T. " }}{PARA 0 "" 0 "" {TEXT -1 18 "t[p] = x-p or 1/x" }}{PARA 0 "" 0 " " {TEXT -1 3 "So:" }}{PARA 0 "" 0 "" {TEXT -1 58 "x = T+p or 1/T wher e T refers to the local parameter t[p]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for p in points do K[p] := convert(subs(x=`if`(p=infi nity,1/T,T+p),K[p]),polynom) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>& %\"KG6#!\"\",,*&,&%\"nGF'*$)F+\"\"#\"\"\"F/F/%\"TG!\"#F/#\"#DF.F'*(\"# dF/F.F'F0F/F'*(\"$l$F/\"\")F'F0F.F'*(\"$X#F/\"\"%F'F0\"\"$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"KG6#\"\"!,.*&,(\"#7!\"\"*$)%\"nG\"\"#\" \"\"F1F/F,F1%\"TG!\"#F1*&F+F1F2F,F1\"#;F1*&\"\")F1F2F1F,*&\"#CF1)F2F0F 1F,*&F+F1)F2\"\"$F1F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"KG6#\"\" \",.*&,(\"\"'!\"\"%\"nGF,*$)F-\"\"#F'F'F'%\"TG!\"#F'*&\"\")F'F1F,F,\" \"&F'*&F5F'F1F'F'*(\"#RF'F0F,F1F0F,*&\"#NF')F1\"\"$F'F'" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%\"KG6#%)infinityG,**&,(\"#7!\"\"*$)%\"nG\"\"# \"\"\"F1F/F1F1)%\"TGF0F1F1*&\"#;F1)F3\"\"$F1F,*&\"#SF1)F3\"\"%F1F,*&\" #OF1)F3\"\"&F1F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "for p i n points do indicial_equation[p] := tcoeff(K[p],T) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%2indicial_equationG6#!\"\",&%\"nGF'*$)F)\"\"#\" \"\"F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%2indicial_equationG6#\"\" !,(\"#7!\"\"*$)%\"nG\"\"#\"\"\"F/F-F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%2indicial_equationG6#\"\"\",(\"\"'!\"\"%\"nGF**$)F+\"\"#F'F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%2indicial_equationG6#%)infinityG,( \"#7!\"\"*$)%\"nG\"\"#\"\"\"F/F-F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "for p in points do exponents[p] := \{solve(indicial_e quation[p],n)\} od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*exponentsG6 #!\"\"<$\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*exponentsG6 #\"\"!<$!\"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*exponentsG6# \"\"\"<$!\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*exponentsG6#% )infinityG<$!\"%\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "fo r p in points do integer_exponents[p] := select(type,exponents[p],inte ger) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%2integer_exponentsG6#! \"\"<$\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%2integer_expon entsG6#\"\"!<$!\"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%2integer _exponentsG6#\"\"\"<$!\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%2 integer_exponentsG6#%)infinityG<$!\"%\"\"$" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 70 "for p in points do min_int_exp[p] := min(op(integer _exponents[p])) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%,min_int_exp G6#!\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%,min_int_expG6#\" \"!!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%,min_int_expG6#\"\"\"!\" #" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%,min_int_expG6#%)infinityG!\"% " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Above you can see how to comp ute:" }}{PARA 0 "" 0 "" {TEXT -1 48 "1) the indicial equation at each \+ of these points" }}{PARA 0 "" 0 "" {TEXT -1 17 "2) the exponents" }} {PARA 0 "" 0 "" {TEXT -1 35 "3) those exponents that are integer" }} {PARA 0 "" 0 "" {TEXT -1 32 "4) the smallest integer exponent" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "We note t hat if L has a nonzero rational solution y in " }{TEXT 260 1 "C" } {TEXT -1 134 "(x), then the valuation of y at x=p, the number v_p(y), \+ is always an integer. This integer must be an element of the set expon ents[p]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 476 "So if the set exponents[p] is empty for any point p, or if it doe s not contain any integers (i.e. if integer_exponents[p] is empty) the n such rational solution y can not exist. So when you compute rational solutions, and you find any point p for which integer_exponents[p]=\{ \} then there are no nonzero rational solutions. If integer_exponents[ p] <> \{\} for all p, then there may (or may not) be rational solution s, but in any case, for any nonzero rational solution y you have:" }} {PARA 0 "" 0 "" {TEXT -1 2 " " }{TEXT 261 30 "v_p(y) in integer_expon ents[p]" }}{PARA 0 "" 0 "" {TEXT -1 10 "Therefore:" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{TEXT 262 24 "v_p(y) >= min_int_exp[p]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "Note that min_int_ex p[p] can also be computed by a Maple command in the DEtools package." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "with(DEtools):\n\n_Envdi ffopdomain:=[Dx,x];\n# This tells Maple that x = our variable, and tha t Dx = differentiation." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%1_Envdiff opdomainG7$%#DxG%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "f or p in points do\n min_int_exp[p]:=gen_exp(L,T,x=p,'restrict_to'=\{' minimal',integer\});\n min_int_exp[p]:=min_int_exp[p][1][1]\nod;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%,min_int_expG6#!\"\"7#7$\"\"!/%\"TG ,&%\"xG\"\"\"F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%,min_int_expG6 #!\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%,min_int_expG6#\"\"! 7#7$!\"$/%\"TG%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%,min_int_exp G6#\"\"!!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%,min_int_expG6#\"\" \"7#7$!\"#/%\"TG,&%\"xGF'F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>& %,min_int_expG6#\"\"\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%,min_ int_expG6#%)infinityG7#7$!\"%/%\"TG*&\"\"\"F.%\"xG!\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%,min_int_expG6#%)infinityG!\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 263 41 "Singularities of a differential operator ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Cons ider again a differential operator L in " }{TEXT 264 1 "C" }{TEXT -1 35 "(x)[Dx]. A point x=p is considered " }{TEXT 265 12 "non-singular" }{TEXT -1 3 " if" }}{PARA 0 "" 0 "" {TEXT -1 68 " *) None of the deno minators of the coefficients of L vanish at x=p" }}{PARA 0 "" 0 "" {TEXT -1 41 " *) lcoeff(L,Dx) does not vanish at x=p." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "All other points, an d the point x=infinity, are considered " }{TEXT 266 17 "singular point s. " }{TEXT -1 52 "There are always only finitely many singular points ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 27 "If p is non-singular, then:" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{TEXT 267 76 "exponents of L at p = \{0,1,2,..,N-1\} where N=degree(L,Dx) is the order of L." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Why? Well, take t=x-p. Then take L(t^n). This will look l ike:" }}{PARA 0 "" 0 "" {TEXT -1 71 "lcoeff(L,Dx) * n*(n-1)*...*(n-(N- 1)) * t^(n-N) plus higher powers of t." }}{PARA 0 "" 0 "" {TEXT -1 32 "So the indicial equation equals:" }}{PARA 0 "" 0 "" {TEXT -1 41 " sub s(x=p, lcoeff(L,Dx)) * n * (n-1)* ..." }}{PARA 0 "" 0 "" {TEXT -1 35 " and the solutions are: n=0, n=1,..." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 95 "In our example, the singular points are \+ -1, 0, 1, infinity. All other poins p are non-singular." }}{PARA 0 "" 0 "" {TEXT -1 95 "Now lets try to compute the rational solutions of L. If y is a non-zero rational solution then:" }}{PARA 0 "" 0 "" {TEXT -1 27 " vp(y) is an element of:" }}{PARA 0 "" 0 "" {TEXT -1 47 "*) \+ \{0,..,N-1\} = \{0,1\} for a non-singular points" }}{PARA 0 "" 0 "" {TEXT -1 73 "*) The set integer_exponents[p], computed above, for all \+ singular points." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "So: vp(y) >=" }}{PARA 0 "" 0 "" {TEXT -1 31 "0 for all no n-singular points p" }}{PARA 0 "" 0 "" {TEXT -1 10 "0 for p=-1" }} {PARA 0 "" 0 "" {TEXT -1 10 "-3 for p=0" }}{PARA 0 "" 0 "" {TEXT -1 10 "-2 for p=1" }}{PARA 0 "" 0 "" {TEXT -1 17 "-4 for p=infinity" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "Therefore , if we write y= (x-1)^0*(x-0)^(-3)*(x-1)^(-2) * P for some P in " } {TEXT 268 1 "C" }{TEXT -1 10 "(x), then " }}{PARA 0 "" 0 "" {TEXT -1 63 "vp(P) >= 0 for all finite p, in other words: P is a polynomial." } }{PARA 0 "" 0 "" {TEXT -1 32 "Furthermore: if p=infinity then:" }} {PARA 0 "" 0 "" {TEXT -1 108 "vp(P) = vp(y / (x-1)^0*(x-0)^(-3)*(x-1)^ (-2)) = vp(y) - vp( (x-1)^0*(x-0)^(-3)*(x-1)^(-2) ) = vp(y)-5 >= -9." }}{PARA 0 "" 0 "" {TEXT -1 92 "So P is a polynomial with valuation >= \+ -9 at infinity, in other words: degree(P,x) <= 9. So:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "Y := P*mul( (x-p)^min_int_exp[p], p =\{op(points)\} minus \{infinity\});" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"YG*(%\"PG\"\"\"%\"xG!\"$,&F(F'F'!\"\"!\"#" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 53 "v_inf_Y:=degree(denom(Y),x)-degree(numer(Y),x) -deg_P;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(v_inf_YG,&\"\"&\"\"\"%&d eg_PG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "v_inf_Y >= mi n_int_exp[infinity];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#1!\"%,&\"\"&\" \"\"%°_PG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "solve( %,deg_P);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*RealRangeG6$,$%)infini tyG!\"\"\"\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "deg_P:=sol ve(v_inf_Y = min_int_exp[infinity], deg_P);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%°_PG\"\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "P:=add(c[i]*x^i,i=0..deg_P);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"PG,6&%\"cG6#\"\"!\"\"\"*&&F'6#F*F*%\"xGF*F**&&F'6#\"\"#F*)F.F2F*F* *&&F'6#\"\"$F*)F.F7F*F**&&F'6#\"\"%F*)F.F " 0 "" {MPLTEXT 1 0 2 "L; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%#DxG\"\"#\"\"\"F(*,\"\"%F(,, *&\"\"$F()%\"xGF*F(F(*&F'F()F/F-F(!\"\"*&\"\")F()F/F'F(F(*&\"\"*F(F/F( F2F-F(F(,&F/F(F(F2!\"#,&*$F5F(F(F(F(F2F/F9F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "Y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,6&%\"cG6 #\"\"!\"\"\"*&&F&6#F)F)%\"xGF)F)*&&F&6#\"\"#F))F-F1F)F)*&&F&6#\"\"$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&6#\"\"(F))F-FJF)F)*&&F&6#\"\")F))F-FOF)F)*&&F&6#\"\"*F))F- FTF)F)F)F-!\"$,&F-F)F)!\"\"!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "normal(apply_L(L,Y,Dx,x));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# ,$*,\"\"#\"\"\",^o*(\"#:F&&%\"cG6#\"\"'F&)%\"x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oF&F&*(F%F&FWF& F@F&F&*&F=F&FUF&FA*&FRF&FPF&FA*(F%F&F*F&F;F&F&*(FCF&F8F&F5F&F&*(FGF&F2 F&FJF&F&*(F7F&FHF&FeoF&F&F&F/!\"%,&*$F@F&F&F&F&FA,&F/F&F&FA!\"$FA" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "solve(\{coeffs(numer(%),x)\} );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<,/&%\"cG6#\"\"*\"\"!/&F&6#\"\" \"F)/&F&6#F)&F&6#\"\"#/&F&6#\"\"%F)/&F&6#\"\"&F)/&F&6#\"\"(F)/&F&6#\" \")F)/&F&6#\"\"'F)/&F&6#\"\"$F)/F1F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Y:=subs(%,Y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" YG*(,&&%\"cG6#\"\"#\"\"\"*&F'F+)%\"xGF*F+F+F+F.!\"$,&F.F+F+!\"\"!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "diffop2de(L,y(x));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*.\"\"%\"\"\",,*&\"\"$F&)%\"xGF%F&F& *&\"\"#F&)F+F)F&!\"\"*&\"\")F&)F+F-F&F&*&\"\"*F&F+F&F/F)F&F&,&F+F&F&F/ !\"#,&*$F2F&F&F&F&F/F+F6-%\"yG6#F+F&F/-%%diffG6$F9-%\"$G6$F+F-F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "ratsols(%,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#*(,&*$)%\"xG\"\"#\"\"\"F*F*F*F*,&F(F*F*!\" \"!\"#F(!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "factor(op(% ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,&*$)%\"xG\"\"#\"\"\"F)F)F)F) ,&F'F)F)!\"\"!\"#F'!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "One mo re example:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "L:=Dx^3+6*x *(2*x^2+3)/(x^2-2)/(x^2+1)*Dx^2+3*(x^2+2)*(11*x^4-5*x^2-2)/x^2/(x^2+1) \n/(x^2-2)^2*Dx+3*(x^2+2)*(5*x^4+5*x^2+2)/x^3/(x^2+1)/(x^2-2)^2;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG,**$)%#DxG\"\"$\"\"\"F**.\"\"'F* %\"xGF*,&*&\"\"#F*)F-F0F*F*F)F*F*,&*$F1F*F*F0!\"\"F4,&F3F*F*F*F4F(F0F* *0F)F*,&F3F*F0F*F*,(*&\"#6F*)F-\"\"%F*F**&\"\"&F*F1F*F4F0F4F*F-!\"#F5F 4F2F?F(F*F**.F)F*F7F*,(*&F>F*F;F*F**&F>F*F1F*F*F0F*F*F-!\"$F5F4F2F?F* " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "factors(numer(lcoeff(L, Dx))*denom(L));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"\"7%7$,&*$)% \"xG\"\"#F$F$F+!\"\"F+7$F*\"\"$7$,&F(F$F$F$F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "[seq(i[1],i=%[2])];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,&*$)%\"xG\"\"#\"\"\"F)F(!\"\"F',&F%F)F)F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "singular_points:=[seq([RootO f(i,x),\"is a root of\",i],i=%),[infinity]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0singular_pointsG7&7%-%'RootOfG6#,&*$)%#_ZG\"\"#\"\" \"F/F.!\"\"Q-is~a~root~of6\",&*$)%\"xGF.F/F/F.F07%\"\"!F1F67%-F(6#,&F+ F/F/F/F1,&F4F/F/F/7#%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "for p in singular_points do\n min_int_exp[p]:=gen_exp(L,T,x= p[1],'restrict_to'=\{'minimal',integer\});\n min_int_exp[p]:=min_int_ exp[p][1][1]\nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%,min_int_expG6 #7%-%'RootOfG6#,&*$)%#_ZG\"\"#\"\"\"F0F/!\"\"Q-is~a~root~of6\",&*$)%\" xGF/F0F0F/F17#7$!\"$/%\"TG,&F7F0F(F1" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>&%,min_int_expG6#7%-%'RootOfG6#,&*$)%#_ZG\"\"#\"\"\"F0F/!\"\"Q-is~a ~root~of6\",&*$)%\"xGF/F0F0F/F1!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>&%,min_int_expG6#7%\"\"!Q-is~a~root~of6\"%\"xG7#7$!\"\"/%\"TGF+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%,min_int_expG6#7%\"\"!Q-is~a~root~o f6\"%\"xG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%,min_int_expG6#7% -%'RootOfG6#,&*$)%#_ZG\"\"#\"\"\"F0F0F0Q-is~a~root~of6\",&*$)%\"xGF/F0 F0F0F07#7$\"\"!/%\"TG,&F6F0F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >&%,min_int_expG6#7%-%'RootOfG6#,&*$)%#_ZG\"\"#\"\"\"F0F0F0Q-is~a~root ~of6\",&*$)%\"xGF/F0F0F0F0\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&% ,min_int_expG6#7#%)infinityG7#7$\"\"\"/%\"TG*&F+F+%\"xG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%,min_int_expG6#7#%)infinityG\"\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "P:='P'; Y:=P*mul(i[3]^min_in t_exp[i], i=\{op(singular_points)\} minus \{[infinity]\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"YG*(%\"PG\"\"\"%\"xG!\"\",&*$)F(\"\"#F'F'F-F)!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "deg_P:='deg_P';degree(denom(Y),x)-d egree(numer(Y),x)-deg_P;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%°_PGF $" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"(\"\"\"%°_PG!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "%=min_int_exp[[infinity]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&\"\"(\"\"\"%°_PG!\"\"F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "deg_P:=solve(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%°_PG\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "P:=add(c[i]*x^i,i=0..deg_P);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG,0&%\"cG6#\"\"!\"\"\"*&&F'6#F*F*%\"xGF*F**&&F'6# \"\"#F*)F.F2F*F**&&F'6#\"\"$F*)F.F7F*F**&&F'6#\"\"%F*)F.F " 0 "" {MPLTEXT 1 0 45 "\{coeffs(numer(normal(apply_L(L,Y,Dx,x))),x)\};" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<),&*&\"#X\"\"\"&%\"cG6#F'F'!\"\"*&\"# gF'&F)6#\"\"&F'F',$*&\"\"$F'F.F'F+,$*&\"#7F'&F)6#F3F'F+,$*&F6F'F(F'F', (*&\"#[F'&F)6#\"\"#F'F+*&\"$#>F'&F)6#\"\"'F'F'*&F=F'&F)6#\"\"!F'F+,&*& \"#FF'F7F'F+*&\"#:F'F(F'F+,&*&F3F'F7F'F'*&FNF'F.F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Y:=normal(subs(solve(%),Y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"YG*(,,&%\"cG6#\"\"#!\"\"*&\"\"%\"\"\"&F(6# \"\"'F.F.*&F'F.)%\"xGF*F.F.*&&F(6#F-F.)F4F-F.F.*&F/F.)F4F1F.F.F.F4F+,& *$F3F.F.F*F+!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 146 "3-dimensiona l space of rational solutions (meaning: there are 3 arbitrary constant s). Order(L)=3, therefore every solution is a rational function." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 233 "Assignme nt: Compute the exponents of the following operator at the points p=0 \+ and p=2 like at the top of the worksheet. The exponents at p=infinity \+ are already calculated for you below. Then compute all rational soluti ons like above." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "L:=Dx^3+ (x-4)/x/(x-2)*Dx^2-12/x^2*Dx+12/x^2/(x-2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG,**$)%#DxG\"\"$\"\"\"F***,&%\"xGF*\"\"%!\"\"F*F-F /,&F-F*\"\"#F/F/F(F1F**(\"#7F*F-!\"#F(F*F/*(F3F*F-F4F0F/F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "How to compute the exponents at x=infinit y?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "p:=infinity;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG%)infinityG" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 25 "t:=1/x; # local parameter" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"tG*&\"\"\"F&%\"xG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "series(normal(apply_L(L, t^n, Dx, x)/t^n),x=p);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&,*\"#7\"\"\"*&\"#6F'%\"nGF'F'*&\" \"#F')F*F,F'!\"\"*$)F*\"\"$F'F.F'%\"xG!\"$F'*&,(*&F,F'F*F'F.*&F,F'F-F' F.\"#CF'F'F2!\"%F'*&,(*&\"\"%F'F*F'F.*&F=F'F-F'F.\"#[F'F'F2!\"&F'-%\"O G6#*&F'F'*$)F2\"\"'F'F.F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "convert(%,polynom);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&,*\"#7 \"\"\"*&\"#6F'%\"nGF'F'*&\"\"#F')F*F,F'!\"\"*$)F*\"\"$F'F.F'%\"xG!\"$F '*&,(*&F,F'F*F'F.*&F,F'F-F'F.\"#CF'F'F2!\"%F'*&,(*&\"\"%F'F*F'F.*&F=F' F-F'F.\"#[F'F'F2!\"&F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "s ubs(x=1/T,%); # express it in terms of the local parameter" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&,*\"#7\"\"\"*&\"#6F'%\"nGF'F'*&\"\"#F')F *F,F'!\"\"*$)F*\"\"$F'F.F')%\"TGF1F'F'*&,(*&F,F'F*F'F.*&F,F'F-F'F.\"#C F'F')F3\"\"%F'F'*&,(*&F:F'F*F'F.*&F:F'F-F'F.\"#[F'F')F3\"\"&F'F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "tcoeff(%,T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*\"#7\"\"\"*&\"#6F%%\"nGF%F%*&\"\"#F%)F(F*F%!\"\" *$)F(\"\"$F%F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "ind_eq:=% ; # indicicial equation at x=infinity" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'ind_eqG,*\"#7\"\"\"*&\"#6F'%\"nGF'F'*&\"\"#F')F*F,F'!\"\"*$)F*\" \"$F'F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "solve(%); # expo nents at infinity" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%!\"\"\"\"$!\"%" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "Now compute the exponents of L \+ at x=0 and x=2 in the same way. Then try to compute the rational solut ions.." }}{PARA 0 "" 0 "" {TEXT -1 64 "Note that you can check if you \+ got the right exponents by doing:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "gen_exp(L,T,x=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#7&!\"%!\"\"\"\"$/%\"TG*&\"\"\"F+%\"xGF&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "gen_exp(L,T,x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#7&!\"$\"\"!\"\"%/%\"TG%\"xG" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 17 "gen_exp(L,T,x=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#7&\"\"!\"\"\"\"\"$/%\"TG,&%\"xGF&\"\"#!\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 401 "Now try to compute the exponents \+ without the gen_exp command. Compute the indicial equation as explaine d earlier in this worksheet, for each p in \{0, 2, infinity\} (I alre ady did p=infinity for you) then compute the exponents, and use the ex ponents to write down an ansatz Y (with unknowns c[..] in it). Find e quations for these unknowns, solve, and substitute, and you get all so lutions of L(y) = 0." }}{PARA 0 "" 0 "" {TEXT -1 102 "Solving a Risch \+ differential equation L(y)=f can be done in a very similar way, but we will stop here." }}}}{MARK "67 1 0" 102 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }