{VERSION 3 0 "SGI MIPS UNIX" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 256 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 261 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning " 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 }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 }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 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 14 "Sample test 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 12 "Question 1. " }{TEXT -1 114 "Integrate the following function. Do not use Maple's in t, but use Maple's ratsols function in the DEtools package." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Notes:" }}{PARA 0 "" 0 "" {TEXT -1 61 "1) See the worksheet on the web for week 8 on h ow to do this." }}{PARA 0 "" 0 "" {TEXT -1 135 "2) During the actual t est you can not use the worksheets on the web, you will have to be abl e to do the following question on your own." }}{PARA 0 "" 0 "" {TEXT -1 500 "3) The worksheet on the web treats the more general case f in \+ K(theta) where K=C(x). It handles f by reducing it to a new f that is \+ in K[theta,theta^(-1)], so the new f is a Laurent polynomial in theta \+ (i.e. a polynomial where you may have negative powers of theta). Going from f in K(theta) to new f in K[theta,theta^(-1)] works in the same \+ way as in the logarithmic case, so that was already covered in test 1. So (and because of time constraints) we'll start with f already in K[ theta,theta^(-1)]." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "rest art; f := (x^2-1-x)/x^3*exp(x+1/x)+(2*x-x^2-1+2*x^4)/x^4/exp(x+1/x)+2/ x*(-x+2*x^2-2)/exp(x+1/x)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG ,(*&*&,(*$)%\"xG\"\"#\"\"\"\"\"\"!\"\"F.F+F/F.-%$expG6#,&F+F.*&F-F-F+! \"\"F.F.F-*$)F+\"\"$F-F5F.*&,*F+F,F)F/F/F.*$)F+\"\"%F-F,F-*&)F+\"\"%F- F0\"\"\"F5F.*&,(F+F/F)F,!\"#F.F-*&F+\"\"\")F0\"\"#F-F5F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 18 "Answer question 1:" }}{PARA 0 "" 0 "" {TEXT -1 36 "Integrate each term of f seperately:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 18 "theta:=exp(x+1/x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&thetaG-%$expG6#,&%\"xG\"\"\"*&\"\"\"F,F)!\"\"F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "d:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "term_of_degree_d := coeff(f,theta,d)*theta^d;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%1term_of_degree_dG*&*&,(*$)%\"xG\"\"#\"\"\"\"\"\"!\" \"F-F*F.F--%$expG6#,&F*F-*&F,F,F*!\"\"F-F-F,*$)F*\"\"$F,F4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 116 "Now integrate this term. Suppose the int egral of this term is c(x)*theta^d, so then we have the following cond ition:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "should_be_zero := term_of_degree_d - diff(c(x)*theta^d,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/should_be_zeroG,(*&*&,(*$)%\"xG\"\"#\"\"\"\"\"\"!\" \"F.F+F/F.-%$expG6#,&F+F.*&F-F-F+!\"\"F.F.F-*$)F+\"\"$F-F5F.*&-%%diffG 6$-%\"cG6#F+F+F.F0F-F/*(F=F.,&F.F.*&F-F-*$)F+\"\"#F-F5F/F.F0F-F/" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "should_be_zero:=normal(shoul d_be_zero/theta^d);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/should_be_ze roG,$*&,.*$)%\"xG\"\"#\"\"\"!\"\"\"\"\"F.F*F.*&-%%diffG6$-%\"cG6#F*F*F .)F*\"\"$F,F.*&F3F.F6F,F.*&F3F,F*F.F-F,*$)F*\"\"$F,!\"\"F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 29 "ratsols(should_be_zero,c(x));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7$7\"*&\"\"\"F&%\"xG!\"\"" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 34 "integral_of_term[d]:=%[2]*theta^d;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%1integral_of_termG6#\"\"\"*&-%$expG6#,&% \"xGF'*&\"\"\"F/F-!\"\"F'F/F-F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "d:=-1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 218 "term_of_degree_d := coeff(f ,theta,d)*theta^d;\nshould_be_zero := term_of_degree_d - diff(c(x)*th eta^d,x);\nshould_be_zero:=normal(should_be_zero/theta^d);\nratsols(sh ould_be_zero,c(x));\nintegral_of_term[d]:=%[2]*theta^d;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%1term_of_degree_dG*&,*%\"xG\"\"#*$)F'F(\"\"\"! \"\"F,\"\"\"*$)F'\"\"%F+F(F+*&)F'\"\"%F+-%$expG6#,&F'F-*&F+F+F'!\"\"F- \"\"\"F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/should_be_zeroG,(*&,*% \"xG\"\"#*$)F(F)\"\"\"!\"\"F-\"\"\"*$)F(\"\"%F,F)F,*&)F(\"\"%F,-%$expG 6#,&F(F.*&F,F,F(!\"\"F.\"\"\"F:F.*&-%%diffG6$-%\"cG6#F(F(F,F5F:F-*&*&F @F.,&F.F.*&F,F,*$)F(\"\"#F,F:F-F.F,F5F:F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/should_be_zeroG*&,0%\"xG\"\"#*$)F'F(\"\"\"!\"\"F,\" \"\"*$)F'\"\"%F+F(*&-%%diffG6$-%\"cG6#F'F'F-F/F+F,*&F5F-F/F+F-*&F5F+F* F+F,F+*$)F'\"\"%F+!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7\",$*&,& \"\"\"F(*$)%\"xG\"\"#\"\"\"F,F-*$)F+\"\"#F-!\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%1integral_of_termG6#!\"\",$*&,&\"\"\"F+*$)%\"xG \"\"#\"\"\"F/F0*&)F.\"\"#F0-%$expG6#,&F.F+*&F0F0F.!\"\"F+\"\"\"F9F'" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "d:=-2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 218 "term_of_degree_d := coeff(f,theta,d)*theta^d;\nshould_be_zero := \+ term_of_degree_d - diff(c(x)*theta^d,x);\nshould_be_zero:=normal(shou ld_be_zero/theta^d);\nratsols(should_be_zero,c(x));\nintegral_of_term[ d]:=%[2]*theta^d;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%1term_of_degree _dG,$*&,(%\"xG!\"\"*$)F(\"\"#\"\"\"F,!\"#\"\"\"F-*&F(\"\"\")-%$expG6#, &F(F/*&F-F-F(!\"\"F/\"\"#F-F8F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/ should_be_zeroG,(*&,(%\"xG!\"\"*$)F(\"\"#\"\"\"F,!\"#\"\"\"F-*&F(\"\" \")-%$expG6#,&F(F/*&F-F-F(!\"\"F/\"\"#F-F8F,*&-%%diffG6$-%\"cG6#F(F(F- *$)F3\"\"#F-F8F)*&*&F>F/,&F/F/*&F-F-*$)F(\"\"#F-F8F)F/F-*$)F3\"\"#F-F8 F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/should_be_zeroG*&,.*$)%\"xG\" \"#\"\"\"!\"#*$)F)\"\"$F+\"\"%F)!\"%*&-%%diffG6$-%\"cG6#F)F)\"\"\"F(F+ !\"\"*&F6F9F(F+F*F6F,F+*$)F)\"\"#F+!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7\",$%\"xG!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%1integral _of_termG6#!\"#,$*&%\"xG\"\"\"*$)-%$expG6#,&F*\"\"\"*&F+F+F*!\"\"F2\" \"#F+F4F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "result:=add(in tegral_of_term[d], d=[1, -1, -2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%'resultG,(*&-%$expG6#,&%\"xG\"\"\"*&\"\"\"F.F+!\"\"F,F.F+F/F,*&,&F,F ,*$)F+\"\"#F.F4F.*&)F+\"\"#F.F'\"\"\"F/!\"\"*&F+F.*$)F'\"\"#F.F/!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "normal(diff(result,x)-f); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "OK." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 258 11 "Question 2." }{TEXT -1 98 " Compu te the indicial equation and the exponents of L at the following two p oints: x=0 and x=1.\n" }}{PARA 0 "" 0 "" {TEXT -1 6 "Notes:" }} {PARA 0 "" 0 "" {TEXT -1 33 "1) See the worksheets of week 12." }} {PARA 0 "" 0 "" {TEXT -1 145 "2) Maple's gen_exp command gives you the exponents but not the indicial equation, so you still have to know ho w to compute the indicial equation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "_Envdiffopdomain:=[Dx,x]:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 34 "L:=x^2*(x-1)^3*Dx^3+(x-1)*Dx+2+24;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG,(*()%\"xG\"\"#\"\"\"),&F(\"\"\"!\"\"F-\"\"$ F*)%#DxGF/F*F-*&F,F-F1F-F-\"#EF-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 261 18 "Answer question 2:" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "apply_L := proc(L, Y, Dx, x) add( \+ coeff(L,Dx,i) * diff(Y,[x$i]), i=0..degree(L,Dx)) end;" }}{PARA 7 "" 1 "" {TEXT -1 42 "Warning, `i` in call to `add` is not local" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(apply_LGR6&%\"LG%\"YG%#DxG%\"xG6\"F+F+-%$ addG6$*&-%&coeffG6%9$9&%\"iG\"\"\"-%%diffG6$9%7#-%\"$G6$9'F5F6/F5;\"\" !-%'degreeG6$F3F4F+F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "The lo cal parameter at x=0 is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "t :=x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"tG%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "apply_L(L, t^n, Dx, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,()%\"xG%\"nG\"#E*&*(,&F%\"\"\"!\"\"F+F+F$F+F&F+\"\" \"F%!\"\"F+*()F%\"\"#F-)F*\"\"$F-,(*&*&F$F-)F&F3F-F-*$)F%\"\"$F-F.F+*& *&F$F-)F&F1F-F-*$)F%\"\"$F-F.!\"$*&*&F$F-F&F-F-*$)F%\"\"$F-F.F1F+F+" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 111 "Now compute the coefficient of \+ the lowest power of t. The easiest way to do that is first to get rid \+ of the t^n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "normal(%/t^n) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,<%\"xG\"#E*&%\"nG\"\"\"F%F)\" \"(F(!\"$*&)F%\"\"$\"\"\")F(F.F/F)*&F-F/)F(\"\"#F/F+*&F-F/F(F/F3*&)F%F 3F/F0F/F+*&F6F/F2F/\"\"**&F6F/F(F/!\"'*&F%F/F0F/F.*&F%F/F2F/!\"**$F0F/ !\"\"*$F2F/F.F/F%!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Lowest \+ power of t=x is -1. So multiply by t to make that -1 a 0." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "normal(%*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,<%\"xG\"#E*&%\"nG\"\"\"F$F(\"\"(F'!\"$*&)F$\"\"$\"\"\" )F'F-F.F(*&F,F.)F'\"\"#F.F**&F,F.F'F.F2*&)F$F2F.F/F.F**&F5F.F1F.\"\"** &F5F.F'F.!\"'*&F$F.F/F.F-*&F$F.F1F.!\"**$F/F.!\"\"*$F1F.F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Now the lowest power is t^0=x^0. The get \+ the coefficient we do: subs(t=0,..)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "indicial_equation_at_0_is:=subs(t=0,%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%:indicial_equation_at_0_isG,(%\"nG!\"$*$)F&\" \"$\"\"\"!\"\"*$)F&\"\"#F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "exponents:=solve(%,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*exp onentsG6%\"\"!,&#\"\"$\"\"#\"\"\"*&%\"IGF+-%%sqrtG6#F)\"\"\"#F+F*,&F(F +F,#!\"\"F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Now for x=1. The l ocal parameter is now:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "t: =x-1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"tG,&%\"xG\"\"\"!\"\"F'" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "apply_L(L, t^n, Dx, x);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,(),&%\"xG\"\"\"!\"\"F'%\"nG\"#E*&F$F 'F)F'F'*()F&\"\"#\"\"\")F%\"\"$F/,(*&*&F$F/)F)F1F/F/*$)F%\"\"$F/!\"\"F '*&*&F$F/)F)F.F/F/*$)F%\"\"$F/F9!\"$*&*&F$F/F)F/F/*$)F%\"\"$F/F9F.F'F' " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "normal(%/t^n);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,,\"#E\"\"\"%\"nGF%*&)%\"xG\"\"#\"\"\" )F&\"\"$F+F%*&F(F+)F&F*F+!\"$*&F(F+F&F%F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "Write it in terms of the local parameter, not in terms of x." }}{PARA 0 "" 0 "" {TEXT -1 65 "t=x-1 so x=t+1. The letter t is a lready used, so use the name T." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "subs(x=T+1,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,\"#E\"\"\" %\"nGF%*&),&%\"TGF%F%F%\"\"#\"\"\")F&\"\"$F,F%*&F(F,)F&F+F,!\"$*&F(F,F &F%F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "indicial_equation_ at_1_is := subs(T=0,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%:indicial _equation_at_1_isG,*\"#E\"\"\"%\"nG\"\"$*$)F(F)\"\"\"F'*$)F(\"\"#F,!\" $" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "exponents:=solve(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*exponentsG6%!\"#,&#\"\"&\"\"#\"\" \"*&%\"IGF+-%%sqrtG6#\"\"$\"\"\"#!\"$F*,&F(F+F,#F1F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "Although I didn't ask for x=infinity in the que stion, lets just do that one also:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "t:=1/x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"tG*&\" \"\"F&%\"xG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "normal( apply_L(L, t^n, Dx, x) / t^n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$* &,<%\"xG!#E*&%\"nG\"\"\"F&F*\"\"(F)!\"$*&)F&\"\"$\"\"\")F)F/F0F**&F.F0 )F)\"\"#F0F/*&F.F0F)F0F4*&)F&F4F0F1F0F,*&F7F0F3F0!\"**&F7F0F)F0!\"'*&F &F0F1F0F/*&F&F0F3F0\"\"**$F1F0!\"\"*$F3F0F,F0F&!\"\"F@" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "normal(subs(x=1/T,%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,<*$)%\"TG\"\"#\"\"\"\"#E*&%\"nG\"\"\"F&F)!\" (*&F,F))F'\"\"$F)F1*$)F,F1F)!\"\"*$)F,F(F)!\"$F,!\"#*&F3F)F'F-F1*&F6F) F'F)\"\"**&F,F)F'F)\"\"'*&F3F)F&F)F7*&F6F)F&F)!\"**&F3F)F0F)F-*&F6F)F0 F)F1F)*$)F'\"\"#F)!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "% *T^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,<*$)%\"TG\"\"#\"\"\"\"#E*&% \"nG\"\"\"F%F(!\"(*&F+F()F&\"\"$F(F0*$)F+F0F(!\"\"*$)F+F'F(!\"$F+!\"#* &F2F(F&F,F0*&F5F(F&F(\"\"**&F+F(F&F(\"\"'*&F2F(F%F(F6*&F5F(F%F(!\"**&F 2F(F/F(F,*&F5F(F/F(F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "in dicial_equation:=subs(T=0,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%2in dicial_equationG,(*$)%\"nG\"\"$\"\"\"!\"\"*$)F(\"\"#F*!\"$F(!\"#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "solve(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6%\"\"!!\"#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 12 "Question 3. " }{TEXT -1 336 "Consider the following differential operator L. Compute the ra tional solutions of L. You may not use ratsols or dsolve. Instead, use the exponents of L given below, to find a form for the rational solut ions with unknown coefficients, find equations for the unknown coeffic ients, solve the equations and find all rational solutions of L." }} {PARA 0 "" 0 "" {TEXT -1 42 "Are all solutions of L rational functions ?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "L:=Dx^4-24*(23*x^3-27 *x^2+13*x-2)/(x-1)^2/(4*x-1)/x^2*Dx^2-48*(26*x^3-9*x^2-4*x+1)/(x-1)^2/ (4*x-1)/x^3*Dx+240*(15*x^2-9*x+1)/(x-1)^2/(4*x-1)/x^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG,**$)%#DxG\"\"%\"\"\"\"\"\"*&*&,**$)%\"xG\" \"$F*\"#B*$)F1\"\"#F*!#FF1\"#8!\"#F+F+)F(F6F*F**(),&F1F+!\"\"F+\"\"#F* ,&F1F)F>F+\"\"\")F1\"\"#F*!\"\"!#C*&*&,*F/\"#EF4!\"*F1!\"%F+F+F+F(F+F* *()F=\"\"#F*F@\"\"\")F1\"\"$F*FD!#[*&,(F4\"#:F1FJF+F+F**()F=\"\"#F*F@ \"\"\")F1\"\"$F*FD\"$S#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "You ma y use the following information:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 262 "finite_singularities:=[seq(i[1],i=factors(denom(L))[ 2])];\nsingularities:=[op(map(RootOf,finite_singularities)), infinity] ;\nfor i in singularities do\n \"The smallest integer exponent at x= \", i, \"is\",\n gen_exp(L,T,x=i,'restrict_to'=\{'minimal',integer\} )[1][1]\nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%5finite_singularitie sG7%%\"xG,&F&\"\"\"#!\"\"\"\"%F(,&F&F(F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.singularitiesG7&\"\"!#\"\"\"\"\"%F(%)infinityG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6&QDThe~smallest~integer~exponent~at~x=6 \"\"\"!Q#isF$!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&QDThe~smallest~in teger~exponent~at~x=6\"#\"\"\"\"\"%Q#isF$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&QDThe~smallest~integer~exponent~at~x=6\"\"\"\"Q#isF$!\" &" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&QDThe~smallest~integer~exponent~a t~x=6\"%)infinityGQ#isF$!#:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 262 18 "Answer question 3:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 38 "We can write a rational solution Y as:" } }{PARA 0 "" 0 "" {TEXT -1 129 "Y = (x-1/4)^(smallest exponent at x=1/4 ) * (x-1)^(smallest exponent at x=1) * (x-0)^(smallest exponent at x=0 ) * (a polynomial P)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Y : = (x-1/4)^0 * (x-1)^(-5) * (x-0)^(-5) * P;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"YG*&%\"PG\"\"\"*&),&%\"xG\"\"\"!\"\"F,\"\"&F')F+\" \"&F'!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "degree(denom( Y),x) - (deg_P + degree(numer(Y),x)) = -15;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&\"#5\"\"\"%°_PG!\"\"!#:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "solve(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#D " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "P:=add(c[i]*x^i,i=0..%) ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"PG,V*&&%\"cG6#\"\")\"\"\")%\" xGF*\"\"\"F+&F(6#\"\"!F+*&&F(6#\"#8F+)F-F5F.F+*&&F(6#\"#7F+)F-F:F.F+*& &F(6#\"#6F+)F-F?F.F+*&&F(6#\"#5F+)F-FDF.F+*&&F(6#\"\"*F+)F-FIF.F+*&&F( 6#\"\"(F+)F-FNF.F+*&&F(6#\"\"'F+)F-FSF.F+*&&F(6#\"\"&F+)F-FXF.F+*&&F(6 #\"\"%F+)F-FgnF.F+*&&F(6#\"\"$F+)F-F\\oF.F+*&&F(6#\"\"#F+)F-FaoF.F+*&& F(6#F+F+F-F+F+*&&F(6#\"#9F+)F-FioF.F+*&&F(6#\"#:F+)F-F^pF.F+*&&F(6#\"# ;F+)F-FcpF.F+*&&F(6#\"# F+)F-FbqF.F+*&&F(6#\"#?F+)F-FgqF.F+*&&F(6#\"#@F+)F-F\\rF.F+*&&F(6#\"#A F+)F-FarF.F+*&&F(6#\"#BF+)F-FfrF.F+*&&F(6#\"#CF+)F-F[sF.F+*&&F(6#\"#DF +)F-F`sF.F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "normal(apply _L(L, Y, Dx, x));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*&,hy*&&%\"cG6# \"#@\"\"\")%\"xG\"#?\"\"\"\"%s8*&&F(6#\"#AF+)F-F*F/\"%q=*&&F(6#\"#BF+) F-F4F/\"%%[#*&&F(6#\"#CF+)F-F:F/\"%IK*&&F(6#\"#DF+)F-F@F/\"%DT*&F-F+&F (6#\"\"!F+\"$D&*&&F(6#\"#8F+)F-\"#9F/!%!\\\"*&&F(6#\"#7F+)F-FQF/!%]5*& &F(6#\"#6F+)F-FXF/!$I'*&&F(6#\"#5F+)F-FhnF/!$v#*&&F(6#\"\"*F+)F-F^oF/! #:*&&F(6#\"\"(F+)F-\"\")F/\"$v\"*&&F(6#\"\"'F+)F-FjoF/\"$?\"*&&F(6#\" \"%F+)F-\"\"&F/!$S\"*&&F(6#\"\"$F+)F-FgpF/!$S#*&&F(6#\"\"#F+)F-F^qF/!$ D#*&&F(6#F+F+)F-FdqF/!\"&*&&F(6#FSF+)F-\"#:F/!%!*=*&&F(6#F`rF+)F-\"#;F /!%v@*&&F(6#FfrF+)F-\"#F/!%l8*&&F(6#FhsF+F,F/Fjp*&&F(6#F.F+F5F/\"%+=*&F'F/F;F/\"%?Y*&F2F /FAF/\"%+&)*&F8F/FGF/\"&NO\"*&F>F/)F-FFF/\"&N-#*&FDF/)F-\"#EF/\"&D&G*& FeoF/&F(6#F\\pF+!$v$*&FjqF/FJF/!%&G\"*&F_rF/FOF/\"$!***&FRF/FVF/\"$!\\ *&FYF/FfnF/\"#!**&FinF/F\\oF/!$!>*&F_oF/FboF/!$V$*&)F-FdoF/FhoF/!$0$*& F[pF/F_pF/!$l\"*&)F-FapF/FepF/\"$K\"*&FhpF/F\\qF/\"$g\"*&FeqF/FhqF/!$X %*&FerF/F]rF/\"%d:*&F[sF/FcrF/\"%X@*&FasF/FirF/\"%&p#*&FgsF/F_sF/\"%NJ *&F,F/FesF/\"%!Q$*&F5F/F[tF/\"%KL*&F;F/F^tF/\"%!)G*&FAF/F'F/\"%+>*&FGF /F2F/\"$b#*&FhtF/F8F/!%0A*&F[uF/F>F/!%Vc*&)F-\"#FF/FDF/!&N-\"*&FeqF/FJ F/\"%+5*&FOF/FerF/!$W\"*&FfnF/FRF/\"#)**&F\\oF/FYF/\"$]\"*&FboF/FinF/F gv*&FhoF/FeoF/Fiu*&F_pF/F_vF/\"#Q*&F\\qF/FdvF/\"#m*&FbqF/FhpF/\"$I#*&F hqF/F_qF/\"$G&*&F]rF/F[sF/!$I$*&FcrF/FasF/!$]&*&FirF/FgsF/!$#z*&F_sF/F ,F/!%S5*&FesF/F5F/!%u7*&F[tF/F;F/!%q9*&F^tF/FAF/!%+;*&F'F/FGF/!%K;*&F2 F/FhtF/!%I:*&F8F/F[uF/!%a7*&F>F/FaxF/!$g(*&F_oF/F_uF/\"$O\"*&F_uF/F_vF /\"$N\"*&F\\oF/F_vF/!#D*&FboF/F[pF/Ffo*&F_pF/FhpF/Fdq*&FepF/FeqF/F[r*& F\\qF/FjqF/!#6*&FbqF/F-F/Ffo*&FcrF/FRF/\"#]*&FirF/F_rF/Fev*&F_sF/FerF/ F[x*&FesF/F[sF/\"$H%*&F[tF/FasF/\"$l'*&F^tF/FgsF/\"$v**&F_uF/F[pF/FipF J!#lFhq!#8*&FOF/FYF/\"$!R*&FVF/FinF/\"$]$*&FfnF/F_oF/\"$l#*&F\\oF/FeoF /\"$l\"*&FboF/F_vF/\"#t*&FhoF/FbpF/!#I*&F_pF/FdvF/F`]l*&FepF/F_qF/\"#[ *&F\\qF/FeqF/\"#&**&FbqF/FjqF/\"$:\"*&FhqF/F-F/\"#v*&F]rF/FRF/\"$[$*&F crF/F_rF/\"$!=*&FirF/FerF/Fbv*&F_sF/F[sF/!$X(*&FesF/FasF/!%D;*&F[tF/Fg sF/!%xG*&F^tF/F,F/!%!e%*&F'F/F5F/!%?o*&F2F/F;F/!%!p**&F8F/FAF/!&!H8*&F >F/FGF/!&Fx\"*&FDF/FhtF/!&:J#*&F_uF/FbpF/!\"'*&FOF/FinF/!#E*&FVF/F_oF/ !#N*&FfnF/FeoF/!#LF/*()F-\"\")F/),&F-F+!\"\"F+\"\")F/,&F-FgpF^`lF+\"\" \"!\"\"F@" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "\{coeffs(numer (expand(%)),x)\}; # sometimes expand is necessary before coeffs is cal led." }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<>,(&%\"cG6#\"\"'\"%!)G&F&6#\" \")!$W\"&F&6#\"\"(!$?(,*&F&6#\"#5\"%+O&F&6#\"#6\"%g@&F&6#\"#7!&+_#&F&6 #\"#8\"%g$*,*F?\"&gP#&F&6#\"#;\"%oJ&F&6#\"#:\"%?V&F&6#\"#9!&g`%,*&F&6# \"\"\"!$?\"&F&6#\"\"$!$k#&F&6#\"\"#\"%gF&F&6#\"\"!!&S3$,*&F&6#\"\"%!%g LFZ\"%?bFV\"%SQF%\"#[,*&F&6#\"#C\"'Sc[&F&6#\"#D!'gZb&F&6#\"#A!&?n$&F&6 #\"#B!&?H&,*F*\"$?\"F.\"%+UF%!%gR&F&6#\"\"*!$g$,,F3!$+'Fip\"%_!&!GN,,&F&6#\"#=\"&?6)F^s\" &GH$FbtF`oF^r!&g\\#Fds!'?*4\",,F]p\"&!)[%F^s!'!oj\"Fds\"&+K%Fgt!&w0$Fb t\"&o*z,,FI\"&![^Fgt\"&'H5FE!&?T&F^r!&!)y\"FM!%?z,,F^r!&+'[FE\"&!okFgt !&+!RFbt\"&gf\"FI!&+K\",,Fbt!&[!pFgt!&gF$Fds\"&+M#F^r\"&S_(FE!&3!>,,F7 !$#zF3\"%gRF.F:FipF\\qF*!%+!*,(FZF\\qFhn\"&+E\"FR\"%+=,,F7\"%_BFI\"%+7 F;\"&g<\"F?!&gd$FM\"%_$),,F3!%+mF7\"%gjF*\"%kKF;!$S)Fip!%K#),,FipFboF3 !%gXF;\"%+%)F7!&?^\"F?!$C',(FV!%gdF]o\"%_6FR\"&sE\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Y:=normal(subs(solve(%),Y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"YG*&,:*&%\"xG\"\"\"&%\"cG6#\"\"!F)!\"&*&&F+ 6#\"\"(F))F(\"\")\"\"\"F.*&)F(\"\"#F5F*F5\"#5*&)F(\"\"*F5F0F5F9*&)F(\" \"$F5F*F5!#5*&F0F5)F(F9F5F@*&)F(\"\"%F5F*F5\"\"&*&)F(\"#6F5F0F5FF*&)F( \"#7F5F0F5!\"\"F*F)*&F0F5)F(F2F5F)*&&F+6#FFF))F(FFF5F)F5*&),&F(F)FMF) \"\"&F5)F(\"\"&F5!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "i ndets(Y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&&%\"cG6#\"\"&&F%6#\"\"( &F%6#\"\"!%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "% minus \+ \{x\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%&%\"cG6#\"\"&&F%6#\"\"(&F% 6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "nops(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 211 "There are 3 indeterminates left. So we find a 3-dimensio nal space of rational solutions. However, L has order 4, therefore it \+ has a 4-dimensional solution space. So not all solutions of L are rati onal functions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Check answer:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "normal(apply_L(L, Y, Dx, x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# \"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "OK." }}}}{MARK "61 0 0" 86 }{VIEWOPTS 1 1 0 1 1 1803 }