{VERSION 6 0 "IBM INTEL NT" "6.0" } {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 "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 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 "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 259 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 260 "Times " 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 261 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 263 "Times " 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 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 "Heading 1" -1 3 1 {CSTYLE " " -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "T ext Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 2 1 3 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 "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 38 "More two point boundary value pro blems" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., C anada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 25.3.2007" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restar t;" }}}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 17 ": The procedur es " }{TEXT 0 9 "desolveFD" }{TEXT -1 5 " and " }{TEXT 0 14 "comparewi thfcn" }{TEXT -1 28 " are used in this worksheet." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "load " }{TEXT 0 7 "desolve" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 263 7 "DEsol.m" }{TEXT -1 32 " is required \+ by this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 121 "It can be read in to a Maple session by a command similar to the one that follows, where the file path gives its location." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "read \"K:\\\\Maple/procdrs/DEsol.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "Error funct ion examples" }}{PARA 0 "" 0 "" {TEXT -1 33 "The error function is def ined by " }{XPPEDIT 18 0 "erf(x) = 2/sqrt(Pi);" "6#/-%$erfG6#%\"xG*&\" \"#\"\"\"-%%sqrtG6#%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp (-x^2),x);" "6#-%$IntG6$-%$expG6#,$*$%\"xG\"\"#!\"\"F+" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Int(exp(-x^2),x)=int(exp(-x^2),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%$expG6#,$*$)%\"xG\"\"#\"\"\"!\"\"F-,$*&#F/F .F/*&%#PiGF3-%$erfG6#F-F/F/F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Diff(erf(x),x)=diff(erf(x),x );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%$erfG6#%\"xGF*,$*( \"\"#\"\"\"%#PiG#!\"\"F--%$expG6#,$*$)F*F-F.F1F.F." }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "plot(erf(x ),x=-3..3,title=\"the error function\");" }}{PARA 13 "" 1 "" {GLPLOT2D 570 250 250 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$!\"$\"\"!$!3g8+. &4z(****!#=7$$!3!******\\2<#pG!#<$!3Wew%fQ/&****F-7$$!3#)***\\7bBav#F1 $!3-If&z5D!****F-7$$!36++]K3XFEF1$!3e5Y'[>uz***F-7$$!3%)****\\F)H')\\# F1$!3ivE7J0!f***F-7$$!3#****\\i3@/P#F1$!3mcrPc^)>***F-7$$!3;++Dr^b^AF1 $!3?HPd@N[&)**F-7$$!3$****\\7Sw%G@F1$!39zKRnb)Q(**F-7$$!3*****\\7;)=,? F1$!3;XjeCwY`**F-7$$!3/++DO\"3V(=F1$!3v9>(R=o'>**F-7$$!3#******\\V'zV< F1$!3I=J[AqSj)*F-7$$!3******\\d;%)G;F1$!3?oc)oc4vy*F-7$$!3!******\\!)H %*\\\"F1$!3!4zH*\\FPg'*F-7$$!3/+++vl[p8F1$!3ok!H\"*fOAZ*F-7$$!3\"***** *\\>iUC\"F1$!3&>(fn4DL:#*F-7$$!3-++DhkaI6F1$!3(H9mfQD9!*)F-7$$!3s***** *\\XF`**F-$!3()z(y\\2@vS)F-7$$!3u*******>#z2))F-$!3/0w!zgB4(yF-7$$!3S+ +]7RKvuF-$!35=KuZ>m&4(F-7$$!3s,+++P'eH'F-$!3'*RxG2`KniF-7$$!3q)***\\7* 3=+&F-$!3Ggv8v#)e1_F-7$$!3[)***\\PFcpPF-$!3#f\"[9#RL.1%F-7$$!3;)****\\ 7VQ[#F-$!3!y')p6[Ihu#F-7$$!32)***\\i6:.8F-$!3eyTjCr;i9F-7$$!3Wb+++v`hH !#?$!3Z'\\,yWFZ1z!)f6X\"F-7$$\"38*** *\\7:xWCF-$\"3Z')Qe:1k/FF-7$$\"3E,++vuY)o$F-$\"3St!Q`'oq!)RF-7$$\"3!z* *****4FL(\\F-$\"33\\\\&f'y_\"=&F-7$$\"3A)****\\d6.B'F-$\"3&fq4aAet@'F- 7$$\"3s****\\(o3lW(F-$\"3CR=k(4Fq2(F-7$$\"35*****\\A))oz)F-$\"39iL(e7a _'yF-7$$\"3e******Hk-,5F1$\"3w2&zaMk7V)F-7$$\"36+++D-eI6F1$\"3@Y492:`, *)F-7$$\"3u***\\(=_(zC\"F1$\"34&[!Q7,?C#*F-7$$\"3M+++b*=jP\"F1$\"3KTs< DO%R[*F-7$$\"3g***\\(3/3(\\\"F1$\"37l@UXRcd'*F-7$$\"33++vB4JB;F1$\"3uV -GeV2$y*F-7$$\"3u*****\\KCnu\"F1$\"3is=BL!y\\')*F-7$$\"3s***\\(=n#f(=F 1$\"3%\\(Ha74@?**F-7$$\"3P+++!)RO+?F1$\"3OFi9>yH`**F-7$$\"30++]_!>w7#F 1$\"3?KO1E6yt**F-7$$\"3O++v)Q?QD#F1$\"3/e30lLk&)**F-7$$\"3G+++5jypBF1$ \"3'oW.k7f>***F-7$$\"3<++]Ujp-DF1$\"3_zQa>)))f***F-7$$\"3++++gEd@EF1$ \"3)Q!=8Xl!z***F-7$$\"39++v3'>$[FF1$\"3_6.P$)Q)*)***F-7$$\"37++D6EjpGF 1$\"3%QN+7j0&****F-7$$\"\"$F*$\"3g8+.&4z(****F--%'COLOURG6&%$RGBG$\"#5 !\"\"$F*F*Fa[l-%+AXESLABELSG6$Q\"x6\"Q!Ff[l-%&TITLEG6#Q3the~error~func tionFf[l-%%VIEWG6$;F(Ffz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "exp(-x^2); " "6#-%$expG6#,$*$%\"xG\"\"#!\"\"" }{TEXT -1 60 "arises in the general solution of the differential equation " }{XPPEDIT 18 0 "dy/dx+2*x*y = 0;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*(\"\"#F'%\"xGF'%\"yGF'F'\"\"!" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "de := diff(y(x),x) + x*y(x) = 0;\neq := dsolve(d e,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6 #%\"xGF-\"\"\"*&F-F.F*F.F.\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# eqG/-%\"yG6#%\"xG*&%$_C1G\"\"\"-%$expG6#,$*$)F)\"\"#F,#!\"\"F3F," }}} {PARA 0 "" 0 "" {TEXT -1 13 "Substituting " }{XPPEDIT 18 0 "dy/dx;" "6 #*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 5 " for " }{TEXT 265 1 "y" }{TEXT -1 43 " in the last differential equation gives " }{XPPEDIT 18 0 "d^ 2*y/(d*x^2)+2*x;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F( *&F&F(F+F(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 0;" "6#/*&%#dyG \"\"\"%#dxG!\"\"\"\"!" }{TEXT -1 70 ", whose general solution is obtai ned from the indefinite integral of " }{XPPEDIT 18 0 "exp(-x^2);" "6# -%$expG6#,$*$%\"xG\"\"#!\"\"" }{TEXT -1 16 "with respect to " }{TEXT 262 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "de := diff(y(x),x$2)+2*x*diff(y(x), x)=0;\neq := dsolve(de,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#de G/,&-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*(F1F2F-F2-F(6$F*F-F2F 2\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG/-%\"yG6#%\"xG,&%$_C1G \"\"\"*&-%$erfGF(F,%$_C2GF,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 1" }}{PARA 256 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "d^2*y/(d*x^2)+2*x;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(* $%\"xGF&F(!\"\"F(*&F&F(F+F(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 0;" "6#/*&%#dyG\"\"\"%#dxG!\"\"\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = 0" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y(1 ) = 1" "6#/-%\"yG6#\"\"\"F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "de := diff(y(x),x $2)+2*x*diff(y(x),x)=0;\nbc := y(0)=0,y(1)=1;\ndesolve(\{de,bc\},y(x), info=true):\ng := unapply(rhs(%),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*(F1F2F-F2-F(6$ F*F-F2F2\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bcG6$/-%\"yG6#\"\" !F*/-F(6#\"\"\"F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%KThe~DE~does~not~have~constant~coefficientsG" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F*\"\"#,$*(F.\"\"\"F*F1-F%6$F'F*F1! \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'%2Substituting~.~.~G/-%#_pG6#%\"xG-%%diffG6$-%\"yGF'F(% &~and~G/-F*6$F%F(-F*6$F,-%\"$G6$F(\"\"#%+gives~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%#_pG6#%\"xGF*,$*(\"\"#\"\"\"F*F.F'F.!\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%\"gGf*6#%\"xG6 \"6$%)operatorG%&arrowGF(*&-%$erfG6#\"\"\"!\"\"-F.6#9$F0F(F(F(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "We can fi nd a discrete numerical solution using " }{TEXT 0 9 "desolveFD" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "soln := desolveFD(\{de,bc\},extrapolate=1,output =points):\ncomparewithfcn(soln,g(x),x);" }}{PARA 6 "" 1 "" {TEXT -1 79 " 0 0 function val: 0 rel \+ err: infinity" }}{PARA 6 "" 1 "" {TEXT -1 81 " .05 .066894 418 function val: .066894416 rel err: 1.8985e-08" }}{PARA 6 " " 1 "" {TEXT -1 81 " .1 .133455337 function val: .13 3455334 rel err: 1.8733e-08" }}{PARA 6 "" 1 "" {TEXT -1 81 " .15 .199354238 function val: .199354234 rel err: 1.805 8e-08" }}{PARA 6 "" 1 "" {TEXT -1 81 " .2 .264272438 fu nction val: .264272433 rel err: 1.7028e-08" }}{PARA 6 "" 1 "" {TEXT -1 81 " .25 .327905701 function val: .327905696 rel err: 1.5553e-08" }}{PARA 6 "" 1 "" {TEXT -1 81 " .3 \+ .3899685 function val: .389968494 rel err: 1.4104e-08" }}{PARA 6 "" 1 "" {TEXT -1 81 " .35 .450197818 function \+ val: .450197812 rel err: 1.2661e-08" }}{PARA 6 "" 1 "" {TEXT -1 81 " .4 .508356422 function val: .508356416 r el err: 1.1213e-08" }}{PARA 6 "" 1 "" {TEXT -1 81 " .45 .5 64235525 function val: .564235519 rel err: 9.2160e-09" }} {PARA 6 "" 1 "" {TEXT -1 81 " .5 .617656808 function va l: .617656803 rel err: 7.4475e-09" }}{PARA 6 "" 1 "" {TEXT -1 81 " .55 .668473762 function val: .668473758 rel \+ err: 5.8342e-09" }}{PARA 6 "" 1 "" {TEXT -1 81 " .6 .7165 72357 function val: .716572354 rel err: 4.1866e-09" }}{PARA 6 "" 1 "" {TEXT -1 81 " .65 .761871039 function val: \+ .761871037 rel err: 2.4939e-09" }}{PARA 6 "" 1 "" {TEXT -1 81 " \+ .7 .80432011 function val: .804320109 rel err: 1. 3676e-09" }}{PARA 6 "" 1 "" {TEXT -1 81 " .75 .843900516 \+ function val: .843900515 rel err: 2.3699e-10" }}{PARA 6 "" 1 " " {TEXT -1 81 " .8 .880622126 function val: .8806221 27 rel err: 5.6778e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " .85 \+ .914521575 function val: .914521576 rel err: 9.8412e-10 " }}{PARA 6 "" 1 "" {TEXT -1 81 " .9 .945659738 functio n val: .945659739 rel err: 8.4597e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " .95 .974118944 function val: .974118945 r el err: 6.1594e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " 1 1 \+ function val: 1 rel err: 4.0000e-10" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 6 "" 1 "" {TEXT -1 48 " \+ Maximum relative error: 1.8985e-08" }}{PARA 6 "" 1 "" {TEXT -1 57 " obtained for the input value: .05 \+ " }}{PARA 6 "" 1 "" {TEXT -1 67 " excluding any cases whe re the function value is zero." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 95 "The points of the discrete solution appea r to lie along the curve for the analytical solution. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "plot([g (x),soln],x=0..1,color=[red,blue],\n style=[line,point],symbol =circle);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 270 270 {PLOTDATA 2 "6'-%' CURVESG6%7S7$$\"\"!F)F(7$$\"3emmm;arz@!#>$\"3Ss$fK.%==HF-7$$\"3[LL$e9u i2%F-$\"3[S\")4rI7baF-7$$\"3nmmm\"z_\"4iF-$\"3:(4O]-/MI)F-7$$\"3[mmmT& phN)F-$\"3)HplG3&H;6!#=7$$\"3CLLe*=)H\\5F?$\"3sNUFJU()*R\"F?7$$\"3gmm \"z/3uC\"F?$\"3!)f')Gn/mh;F?7$$\"3%)***\\7LRDX\"F?$\"3!\\>c-Zi8$>F?7$$ \"3]mm\"zR'ok;F?$\"3P&)))f<1g3AF?7$$\"3w***\\i5`h(=F?$\"3YVF#*[\"3I[#F ?7$$\"3WLLL3En$4#F?$\"3+:;km^+jFF?7$$\"3qmm;/RE&G#F?$\"3BROh%*=`2IF?7$ $\"3\")*****\\K]4]#F?$\"3mq;QDBD!G$F?7$$\"3$******\\PAvr#F?$\"3q,enS![ 6b$F?7$$\"3)******\\nHi#HF?$\"3]&f2]b4#4QF?7$$\"3jmm\"z*ev:JF?$\"3m!oS k$\\%3/%F?7$$\"3?LLL347TLF?$\"3,BzY*eJFJ%F?7$$\"3,LLLLY.KNF?$\"3Yv&o%y W))RXF?7$$\"3w***\\7o7Tv$F?$\"3@'*HZ)p!G+[F?7$$\"3'GLLLQ*o]RF?$\"3a^i$ )H*)=F]F?7$$\"3A++D\"=lj;%F?$\"3u<@k[d6s_F?7$$\"31++vV&Rjgt&F?7$$\"3GLLeR\"3Gy%F?$\"3_s9=) QNw%fF?7$$\"3cmm;/T1&*\\F?$\"3S!\\Ek_>9<'F?7$$\"3&em;zRQb@&F?$\"3-r?Ls e*))R'F?7$$\"3\\***\\(=>Y2aF?$\"3w-i.3zq#f'F?7$$\"39mm;zXu9cF?$\"3Y`W3 Nnb(z'F?7$$\"3l******\\y))GeF?$\"3DMc%=_NU+(F?7$$\"3'*)***\\i_QQgF?$\" 3$pgzi1+:?(F?7$$\"3@***\\7y%3TiF?$\"3)[?#*yVtwQ(F?7$$\"35****\\P![hY'F ?$\"31L5))Ro$*)e(F?7$$\"3kKLL$Qx$omF?$\"3anlWTw&[w(F?7$$\"3!)*****\\P+ V)oF?$\"3M.Z`CJ_ZzF?7$$\"3?mm\"zpe*zqF?$\"3Zu3MvZU3\")F?7$$\"3%)***** \\#\\'QH(F?$\"3AN$eyP-$z#)F?7$$\"3GKLe9S8&\\(F?$\"3UVboS8HN%)F?7$$\"3R ***\\i?=bq(F?$\"3K/&)[]'*Q$f)F?7$$\"3\"HLL$3s?6zF?$\"3Tn=;pO3V()F?7$$ \"3a***\\7`Wl7)F?$\"3;G(RKWkY*))F?7$$\"3#pmmm'*RRL)F?$\"3;H8)Q/Ed.*F?7 $$\"3Qmm;a<.Y&)F?$\"3G([$HUc-v\"*F?7$$\"3=LLe9tOc()F?$\"3'*z7A,TF3$*F? 7$$\"3u******\\Qk\\*)F?$\"3\\LDPIeYE%*F?7$$\"3CLL$3dg6<*F?$\"35&\\g^s( )pb*F?7$$\"3ImmmmxGp$*F?$\"3)fauz(HKp'*F?7$$\"3A++D\"oK0e*F?$\"3Ah))>3 !)e%y*F?7$$\"3A++v=5s#y*F?$\"3O>[umzi!*)*F?7$$\"\"\"F)Fbz-%'COLOURG6&% $RGBG$\"*++++\"!\")F(F(-%&STYLEG6#%%LINEG-F$6%77F'7$$\"3G+++++++]F-$\" 35+++rF?7$$\"35+++++++?F?$\"3#)*****zPCFk#F?7$$\"3++++++++DF ?$\"35+++7q0zKF?7$$\"3))**************HF?$\"3!)*****z*\\o**QF?7$$\"3w* ************\\$F?$\"3!)*****4=y>]%F?7$$\"3A+++++++SF?$\"3]*****p@kN3&F ?7$$\"35+++++++XF?$\"3Q+++Y_NUcF?7$$\"3++++++++]F?$\"3o*****f2ol<'F?7$ $\"3U+++++++bF?$\"3!)*****4iPZo'F?7$$\"3w**************fF?$\"3g*****fc Bd;(F?7$$\"3A+++++++lF?$\"3!)*****4R5(=wF?7$$\"3a**************pF?$\"3 K++++6?V!)F?7$$\"3++++++++vF?$\"3.+++b^+R%)F?7$$\"3U+++++++!)F?$\"3A++ +i7A1))F?7$$\"3w*************\\)F?$\"3#******>v:_9*F?7$$\"3A+++++++!*F ?$\"3]*****RQ(fc%*F?7$$\"3a*************\\*F?$\"3')******R%*=T(*F?Faz- Fez6&FgzF(F(Fhz-F\\[l6#%&POINTG-%+AXESLABELSG6$Q\"x6\"Q!Fjal-%'SYMBOLG 6#%'CIRCLEG-%%VIEWG6$;F(Fbz%(DEFAULTG" 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "We can also obtain a n umerical solution in the form of a procedure." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "de := diff( y(x),x$2)+2*x*diff(y(x),x)=0;\nbc := y(0)=0,y(1)=1;\ngn := desolveFD( \{de,bc\},y(x),output=localtaylor);\nplot('gn'(x),x=0..1,color=coral,t hickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-% \"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*(F1F2F-F2-F(6$F*F-F2F2\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bcG6$/-%\"yG6#\"\"!F*/-F(6#\"\"\"F." }} {PARA 13 "" 1 "" {GLPLOT2D 415 284 284 {PLOTDATA 2 "6'-%'CURVESG6#7S7$ $\"\"!F)F(7$$\"+;arz@!#6$\"+KS==HF-7$$\"+XTFwSF-$\"+qI7baF-7$$\"+\"z_ \"4iF-$\"+CSS.$)F-7$$\"+S&phN)F-$\"+$3&H;6!#57$$\"+*=)H\\5F?$\"+JU()*R \"F?7$$\"+[!3uC\"F?$\"+n/mh;F?7$$\"+J$RDX\"F?$\"+qCOJ>F?7$$\"+)R'ok;F? $\"+=1g3AF?7$$\"+1J:w=F?$\"+\\\"3I[#F?7$$\"+3En$4#F?$\"+m^+jFF?7$$\"+/ RE&G#F?$\"+%*=`2IF?7$$\"+D.&4]#F?$\"+DBD!G$F?7$$\"+vB_jgt&F?7$$\"+R\"3Gy%F?$\"+)QNw%fF?7$$\"+.T1&*\\F?$\" +D&>9<'F?7$$\"+(RQb@&F?$\"+re*))R'F?7$$\"+=>Y2aF?$\"+2zq#f'F?7$$\"+yXu 9cF?$\"+Mnb(z'F?7$$\"+\\y))GeF?$\"+@bB/qF?7$$\"+i_QQgF?$\"+m+],sF?7$$ \"+!y%3TiF?$\"+PMn(Q(F?7$$\"+O![hY'F?$\"+Ro$*)e(F?7$$\"+#Qx$omF?$\"+Sw &[w(F?7$$\"+u.I%)oF?$\"+CJ_ZzF?7$$\"+(pe*zqF?$\"+vZU3\")F?7$$\"+C\\'QH (F?$\"+xBIz#)F?7$$\"+8S8&\\(F?$\"+R8HN%)F?7$$\"+0#=bq(F?$\"+]'*Q$f)F?7 $$\"+2s?6zF?$\"+oO3V()F?7$$\"+IXaE\")F?$\"+UWm%*))F?7$$\"+l*RRL)F?$\"+ VgsN!*F?7$$\"+`<.Y&)F?$\"+Uc-v\"*F?7$$\"+8tOc()F?$\"++TF3$*F?7$$\"+\\Q k\\*)F?$\"+IeYE%*F?7$$\"+p0;r\"*F?$\"+Cx)pb*F?7$$\"+lxGp$*F?$\"+xHKp'* F?7$$\"+!oK0e*F?$\"+3!)e%y*F?7$$\"+<5s#y*F?$\"+mzi!*)*F?7$$\"\"\"F)Fbz -%*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q!F\\[l-%'COLOURG6&%$RGBG$ \"*++++\"!\")$\")AR!)\\Fd[lF(-%%VIEWG6$;F(Fbz%(DEFAULTG" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "The value of the numer ical procedure " }{TEXT 263 2 "gn" }{TEXT -1 143 " which provides a co ntinuous solution for the boundary value problem can be compared with \+ that of the analytial solution at specific values of " }{TEXT 266 1 "x " }{TEXT -1 17 " between 0 and 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "xx := evalf(sqrt(5)/3);\ngn( xx);\nevalf(evalf[13](g(xx)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x xG$\"+B*fNX(!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+v1X.%)!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+v1X.%)!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)-x;" "6#,&*(%\"dG\"\"#%\"yG\"\"\" *&F%F(*$%\"xGF&F(!\"\"F(F+F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx-y \+ = 0;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'%\"yGF)\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "y(0) = 0" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "y(1) = 1" "6#/-%\"yG6#\"\"\"F'" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "The analy tical solution of this boundary value problem given by " }{TEXT 0 6 "d solve" }{TEXT -1 30 " involves the error function. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "de := dif f(y(x),x$2)-x*diff(y(x),x)-y(x)=0;\nbc := y(0)=0,y(1)=1;\ndsolve(\{de, bc\},y(x)):\ng := unapply(rhs(%),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#deG/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&F-F2-F(6$F*F- F2!\"\"F*F6\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bcG6$/-%\"yG6# \"\"!F*/-F(6#\"\"\"F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\" xG6\"6$%)operatorG%&arrowGF(**-%$erfG6#,$*&#\"\"\"\"\"#F3*&F4F29$F3F3F 3F3-%$expG6#,$*&F2F3*$)F6F4F3F3F3F3-F.6#,$*&F4!\"\"F4F2F3FB-F86#F2FBF( F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 " We can find a discrete numerical solution using " }{TEXT 0 9 "desolveF D" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 84 "soln := desolveFD(\{de,bc\},extrapolate=2,ou tput=points):\ncomparewithfcn(soln,g(x),x);" }}{PARA 6 "" 1 "" {TEXT -1 79 " 0 0 function val: 0 r el err: infinity" }}{PARA 6 "" 1 "" {TEXT -1 81 " .05 .035 473296 function val: .035473297 rel err: 5.6380e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " .1 .071124255 function val: \+ .071124255 rel err: 0.0000e-01" }}{PARA 6 "" 1 "" {TEXT -1 81 " \+ .15 .10713232 function val: .10713232 rel err: 9. 3343e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " .2 .143680524 \+ function val: .143680524 rel err: 0.0000e-01" }}{PARA 6 "" 1 " " {TEXT -1 81 " .25 .180957349 function val: .1809573 49 rel err: 1.1052e-09" }}{PARA 6 "" 1 "" {TEXT -1 81 " .3 \+ .219158675 function val: .219158675 rel err: 0.0000e-01 " }}{PARA 6 "" 1 "" {TEXT -1 81 " .35 .258489834 functio n val: .258489834 rel err: 0.0000e-01" }}{PARA 6 "" 1 "" {TEXT -1 81 " .4 .299167813 function val: .299167813 r el err: 0.0000e-01" }}{PARA 6 "" 1 "" {TEXT -1 81 " .45 .3 41423637 function val: .341423637 rel err: 5.8578e-10" }} {PARA 6 "" 1 "" {TEXT -1 81 " .5 .385504972 function va l: .385504972 rel err: 0.0000e-01" }}{PARA 6 "" 1 "" {TEXT -1 81 " .55 .431678991 function val: .431678991 rel \+ err: 4.6331e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " .6 .4802 35548 function val: .480235548 rel err: 0.0000e-01" }}{PARA 6 "" 1 "" {TEXT -1 81 " .65 .53149073 function val: \+ .53149073 rel err: 1.8815e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " \+ .7 .585790833 function val: .585790832 rel err: 3. 4142e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " .75 .643516843 \+ function val: .643516843 rel err: 3.1079e-10" }}{PARA 6 "" 1 " " {TEXT -1 81 " .8 .705089517 function val: .7050895 17 rel err: 2.8365e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " .85 \+ .770975139 function val: .770975139 rel err: 2.5941e-10 " }}{PARA 6 "" 1 "" {TEXT -1 81 " .9 .841692085 functio n val: .841692085 rel err: 1.1881e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " .95 .917818324 function val: .917818324 r el err: 2.1791e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " 1 1 \+ function val: 1 rel err: 0.0000e-01" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 6 "" 1 "" {TEXT -1 48 " \+ Maximum relative error: 1.1052e-09" }}{PARA 6 "" 1 "" {TEXT -1 57 " obtained for the input value: .25 \+ " }}{PARA 6 "" 1 "" {TEXT -1 67 " excluding any cases whe re the function value is zero." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 95 "The points of the discrete solution appea r to lie along the curve for the analytical solution. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "plot([g (x),soln],x=0..1,color=[red,blue],style=[line,point],\n symbol= circle);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 270 270 {PLOTDATA 2 "6'-%'C URVESG6%7S7$$\"\"!F)F(7$$\"3emmm;arz@!#>$\"3m9FU!H!RX:F-7$$\"3[LL$e9ui 2%F-$\"39Qt[<%p6*GF-7$$\"3nmmm\"z_\"4iF-$\"3i`rPHM<2WF-7$$\"3[mmmT&phN )F-$\"3#*p#)Ru^GPfF-7$$\"3CLLe*=)H\\5!#=$\"3M;bS`1dluF-7$$\"3gmm\"z/3u C\"FB$\"3?4l-$))p&))))F-7$$\"3%)***\\7LRDX\"FB$\"3&Hs6#*)3%p.\"FB7$$\" 3]mm\"zR'ok;FB$\"3sV$>ta:5>\"FB7$$\"3w***\\i5`h(=FB$\"3+#eGU)GnX8FB7$$ \"3WLLL3En$4#FB$\"32`(3k/Hg]\"FB7$$\"3qmm;/RE&G#FB$\"3--oxMNY[;FB7$$\" 3\")*****\\K]4]#FB$\"3Z!GIy:!H5=FB7$$\"3$******\\PAvr#FB$\"3y(*z*y$>^u >FB7$$\"3)******\\nHi#HFB$\"3\\`C*fLkX8#FB7$$\"3jmm\"z*ev:JFB$\"3_Nr%Q wg:G#FB7$$\"3?LLL347TLFB$\"3O>B)QQm&eCFB7$$\"3,LLLLY.KNFB$\"3%\\;D6#G` 5EFB7$$\"3w***\\7o7Tv$FB$\"3mbq#R^4)*y#FB7$$\"3'GLLLQ*o]RFB$\"3=)\\k? \\)*3&HFB7$$\"3A++D\"=lj;%FB$\"3KjvacgSIJFB7$$\"31++vV&RY2aFB$\"3*=(R0qdiHUFB7$$\"39mm;zXu9cFB$\"3b%)[O Ra+EWFB7$$\"3l******\\y))GeFB$\"30z^/:3JLYFB7$$\"3'*)***\\i_QQgFB$\"3# 4]^$4fqS[FB7$$\"3@***\\7y%3TiFB$\"3+1rd]t\"f/&FB7$$\"35****\\P![hY'FB$ \"3yVWUD_Gz_FB7$$\"3kKLL$Qx$omFB$\"3sziF8J=%\\&FB7$$\"3!)*****\\P+V)oF B$\"38:voToNHdFB7$$\"3?mm\"zpe*zqFB$\"3o(*Rskn\"y%fFB7$$\"3%)*****\\# \\'QH(FB$\"3a&)*yGjqF>'FB7$$\"3GKLe9S8&\\(FB$\"3U#*yPnBPHkFB7$$\"3R*** \\i?=bq(FB$\"3o\\PmTRM$o'FB7$$\"3\"HLL$3s?6zFB$\"3)*pG!R`T&QpFB7$$\"3a ***\\7`Wl7)FB$\"3<'H #FB7$$\"3w*************\\$FB$\"3/+++P$)*[e#FB7$$\"3A+++++++SFB$\"3y*** **p7y;*HFB7$$\"35+++++++XFB$\"3-+++qjB9MFB7$$\"3++++++++]FB$\"3y*****R s\\]&QFB7$$\"3U+++++++bFB$\"3&******z!**y;VFB7$$\"3w**************fFB$ \"3/+++zaN-[FB7$$\"3A+++++++lFB$\"3J+++,t!\\J&FB7$$\"3a**************p FB$\"3q*****\\K3z&eFB7$$\"3++++++++vFB$\"38+++J%o^V'FB7$$\"3U+++++++!) FB$\"3++++q^*30(FB7$$\"3w*************\\)FB$\"3!******\\Q^(4xFB7$$\"3A +++++++!*FB$\"35+++Z3#pT)FB7$$\"3a*************\\*FB$\"3K+++NK=y\"*FB7 $FbzFbz-Fhz6&FjzF(F(F[[l-F_[l6#%&POINTG-%+AXESLABELSG6$Q\"x6\"Q!F^bl-% 'SYMBOLG6#%'CIRCLEG-%%VIEWG6$;F(Fbz%(DEFAULTG" 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "We can also obtain a n umerical solution in the form of a procedure." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "de := diff( y(x),x$2)-x*diff(y(x),x)-y(x)=0;\nbc := y(0)=0,y(1)=1;\ngn := desolveF D(\{de,bc\},output=localtaylor);\nplot('gn'(x),x=0..1,color=coral,thic kness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG 6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&F-F2-F(6$F*F-F2!\"\"F*F6\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bcG6$/-%\"yG6#\"\"!F*/-F(6#\"\"\"F." }} {PARA 13 "" 1 "" {GLPLOT2D 378 262 262 {PLOTDATA 2 "6'-%'CURVESG6#7S7$ $\"\"!F)F(7$$\"+;arz@!#6$\"+!H!RX:F-7$$\"+XTFwSF-$\"+<%p6*GF-7$$\"+\"z _\"4iF-$\"+HM<2WF-7$$\"+S&phN)F-$\"+t^GPfF-7$$\"+*=)H\\5!#5$\"+\\1dluF -7$$\"+[!3uC\"FB$\"+%))p&))))F-7$$\"+J$RDX\"FB$\"+*)3%p.\"FB7$$\"+)R'o k;FB$\"+Zb,\">\"FB7$$\"+1J:w=FB$\"+%)GnX8FB7$$\"+3En$4#FB$\"+Y!Hg]\"FB 7$$\"+/RE&G#FB$\"+NNY[;FB7$$\"+D.&4]#FB$\"+e,H5=FB7$$\"+vB_^ u>FB7$$\"+v'Hi#HFB$\"+OVcM@FB7$$\"+(*ev:JFB$\"+j2c\"G#FB7$$\"+347TLFB$ \"+%Qm&eCFB7$$\"+LY.KNFB$\"+@G`5EFB7$$\"+\"o7Tv$FB$\"+9&4)*y#FB7$$\"+$ Q*o]RFB$\"+#\\)*3&HFB7$$\"+\"=lj;%FB$\"+cgSIJFB7$$\"+V&RY2aFB$\"+pdiHUFB7$$\"+yX u9cFB$\"+Qa+EWFB7$$\"+\\y))GeFB$\"+93JLYFB7$$\"+i_QQgFB$\"+4fqS[FB7$$ \"+!y%3TiFB$\"+\\t\"f/&FB7$$\"+O![hY'FB$\"+C_Gz_FB7$$\"+#Qx$omFB$\"+7J =%\\&FB7$$\"+u.I%)oFB$\"+ToNHdFB7$$\"+(pe*zqFB$\"+kn\"y%fFB7$$\"+C\\'Q H(FB$\"+K1x#>'FB7$$\"+8S8&\\(FB$\"+lBPHkFB7$$\"+0#=bq(FB$\"+SRM$o'FB7$ $\"+2s?6zFB$\"+K:aQpFB7$$\"+IXaE\")FB$\"+cGP8sFB7$$\"+l*RRL)FB$\"+W/%e [(FB7$$\"+`<.Y&)FB$\"+ALvsxFB7$$\"+8tOc()FB$\"+ms&f1)FB7$$\"+\\Qk\\*)F B$\"+3 " 0 "" {MPLTEXT 1 0 57 "xx := evalf( sqrt(5)/3);\ngn(xx);\nevalf(evalf[15](g(xx)));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"+B*fNX(!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $\"+l'***zj!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+l'***zj!#5" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 256 " " 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+x;" "6#,&*(%\"dG \"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(F+F(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx-x*y = 0;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&%\"xGF '%\"yGF'F)\"\"!" }{TEXT -1 21 ", y(0) = 0, y(1) = 1 " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "The analytical solution of this boundary value problem given by " }{TEXT 0 6 "dsolve" }{TEXT -1 36 " again involves the error function. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "de := diff(y(x),x $2)+x*diff(y(x),x)-x*y(x)=0;\nbc := y(0)=0,y(1)=1;\ndsolve(\{de,bc\},y (x)):\ng := unapply(rhs(%),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#d eG/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&F-F2-F(6$F*F-F2F2*& F-F2F*F2!\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bcG6$/-%\"yG6 #\"\"!F*/-F(6#\"\"\"F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#% \"xG6\"6$%)operatorG%&arrowGF(,&*,-%$expG6##\"\"(\"\"#\"\"\",&-%$erfG6 #*&F3#F4F3^#F4F4F4-F76#*&^##\"\"$F3F4F3F:!\"\"FBF6F4-F/6#!\"#F4-F/6#,$ *&#F4F3F4*&9$F4,&F3F4FLF4F4F4FBF4F4**F.F4F5FB-F76#*(^#F:F4F3F:FMF4F4-F /6#,(FLFB*&#F4F3F4*$)FLF3F4F4FBF3FBF4FBF(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "We can find a discrete nu merical solution using " }{TEXT 0 9 "desolveFD" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "soln := desolveFD(\{de,bc\},extrapolate=2,output=points);\nsoln : = [op(2..nops(soln),soln)]:\ncomparewithfcn(soln,g(x),x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%solnG777$$\"\"!F(F'7$$\"+++++]!#6$\"+&=QxR&F ,7$$\"+++++5!#5$\"+f(y#y5F27$$\"+++++:F2$\"+l%yVh\"F27$$\"+++++?F2$\"+ 0u3Z@F27$$\"+++++DF2$\"+:fdvEF27$$\"+++++IF2$\"+`K<*>$F27$$\"+++++NF2$ \"+_(otr$F27$$\"+++++SF2$\"+&=6)HUF27$$\"+++++XF2$\"+bkIOZF27$$F+F2$\" +zP\"oB&F27$$\"+++++bF2$\"+@0WJdF27$$\"+++++gF2$\"+leV?iF27$$\"+++++lF 2$\"+-N=/nF27$$\"+++++qF2$\"+JN>$=(F27$$\"+++++vF2$\"+hP4ewF27$$\"++++ +!)F2$\"+21iH\")F27$$\"+++++&)F2$\"+X(4')f)F27$$\"+++++!*F2$\"+8n)f1*F 27$$\"+++++&*F2$\"+0wvK&*F27$$\"\"\"F(Fdq" }}{PARA 6 "" 1 "" {TEXT -1 81 " .05 .053977382 function val: .053977382 rel \+ err: 2.7789e-09" }}{PARA 6 "" 1 "" {TEXT -1 81 " .1 .1078 27876 function val: .107827876 rel err: 2.7822e-09" }}{PARA 6 "" 1 "" {TEXT -1 81 " .15 .161437847 function val: \+ .161437846 rel err: 6.1943e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " \+ .2 .214708741 function val: .21470874 rel err: 2. 3287e-09" }}{PARA 6 "" 1 "" {TEXT -1 81 " .25 .267557592 \+ function val: .267557591 rel err: 1.8688e-09" }}{PARA 6 "" 1 " " {TEXT -1 81 " .3 .319917325 function val: .3199173 25 rel err: 1.5629e-09" }}{PARA 6 "" 1 "" {TEXT -1 81 " .35 \+ .371736875 function val: .371736875 rel err: 1.8831e-09 " }}{PARA 6 "" 1 "" {TEXT -1 81 " .4 .422981119 functio n val: .422981118 rel err: 1.4185e-09" }}{PARA 6 "" 1 "" {TEXT -1 81 " .45 .473630646 function val: .473630645 r el err: 1.6891e-09" }}{PARA 6 "" 1 "" {TEXT -1 81 " .5 .5 23681378 function val: .523681377 rel err: 2.1005e-09" }} {PARA 6 "" 1 "" {TEXT -1 81 " .55 .573144052 function va l: .573144053 rel err: 2.0937e-09" }}{PARA 6 "" 1 "" {TEXT -1 81 " .6 .622043587 function val: .622043588 rel \+ err: 2.2506e-09" }}{PARA 6 "" 1 "" {TEXT -1 81 " .65 .6704 1835 function val: .670418351 rel err: 1.6408e-09" }}{PARA 6 "" 1 "" {TEXT -1 81 " .7 .718319353 function val: \+ .718319354 rel err: 1.2529e-09" }}{PARA 6 "" 1 "" {TEXT -1 81 " \+ .75 .765809376 function val: .765809377 rel err: 1. 4364e-09" }}{PARA 6 "" 1 "" {TEXT -1 81 " .8 .812962061 \+ function val: .812962062 rel err: 1.4761e-09" }}{PARA 6 "" 1 " " {TEXT -1 81 " .85 .859860975 function val: .8598609 74 rel err: 1.1630e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " .9 \+ .906598671 function val: .906598672 rel err: 7.7212e-10 " }}{PARA 6 "" 1 "" {TEXT -1 81 " .95 .953275761 functio n val: .95327576 rel err: 1.0490e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " 1 1 function val: 1 r el err: 0.0000e-01" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 6 " " 1 "" {TEXT -1 48 " Maximum relative error: 2.7822e-09" }}{PARA 6 "" 1 "" {TEXT -1 57 " obtained for the input va lue: .1 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 259 4 "Note" }{TEXT -1 92 ": The first point is omitted becaus e the analytical solution evaluates to a non-zero number." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "We can also obtain a numerical solution in the form of a procedure." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 183 "de := diff( y(x),x$2)+x*diff(y(x),x)-x*y(x)=0;\nbc := y(0)=0,y(1)=1;\ngn := desolv eFD(\{de,bc\},output=localtaylor);\nplot(['gn'(x),g(x)],x=-.2..1.2,col or=[red,aquamarine],thickness=[1,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&F-F2-F(6$F*F -F2F2*&F-F2F*F2!\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bcG6$/ -%\"yG6#\"\"!F*/-F(6#\"\"\"F." }}{PARA 13 "" 1 "" {GLPLOT2D 371 269 269 {PLOTDATA 2 "6&-%'CURVESG6%7N7$$\"+NIeJm!#8$\"+fx,irF*7$$\"+pwgg:! #7$\"+\"4Mao\"F07$$\"+Mq0eCF0$\"+^]maEF07$$\")k]bL!#5$\"+zQ*Qi$F07$$\" +0,e?=!#6$\"++Q4m>FA7$$\"*d4cI$F;$\"+nMPpNFA7$$\"*\\VhE'F;$\"+Z71jnFA7 $$\"*_;9J*F;$\"+;kB/5F;7$$\"+m%p$*>\"F;$\"+E%*Q#H\"F;7$$\"+b/L,:F;$\"+ d3!eh\"F;7$$\"+D8`/=F;$\"+Y
F;7$$\"+X:s'4#F;$\"+`@n\\AF;7$$\"+e#e?O #F;$\"+dwBIDF;7$$\"+s#pvn#F;$\"+2a6iGF;7$$\"+([[[%HF;$\"+RgnTJF;7$$\"+ axvbKF;$\"+#)*Q\\Y$F;7$$\"+P^'4`$F;$\"+PVF\\PF;7$$\"+a7\"H$QF;$\"+(o># fSF;7$$\"+i`V?TF;$\"+^)eBN%F;7$$\"+0mV?WF;$\"+O36cYF;7$$\"+'RJfp%F;$\" +Iu9L\\F;7$$\"+Z(*3$*\\F;$\"+,q$*H_F;7$$\"+ePv,`F;$\"+ca,ObF;7$$\"+(oY /d&F;$\"+;,n+eF;7$$\"+7CkgeF;$\"+j\"zF;$ \"+\\kyY!)F;7$$\"*!4T6#)!\"*$\"+yu*o*F;7$$\"+0-)f\")*F;$\"+*e5z#)*F;7$$\"* YWW'**Fhu$\"+HWtm**F;7$$\"+Smkt**F;$\"+$=V`(**F;7$$\"+?)[G)**F;$\"+gC& R)**F;7$$\"++50#***F;$\"+oAc#***F;7$%*undefinedGF^y-%'COLOURG6&%$RGBG$ \"*++++\"!\")$\"\"!FgyFfy-%*THICKNESSG6#\"\"\"-F$6%7S7$$!\"#!\"\"$!+a. 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"6#$\"+@\"RTh(!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" +@\"RTh(!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "Bessel function examples" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 17 "Bessel functions " }{XPPEDIT 18 0 "J[v];" "6#&%\"JG6#%\"vG" } {TEXT -1 8 "(x) and " }{XPPEDIT 18 0 "Y[v];" "6#&%\"YG6#%\"vG" }{TEXT -1 39 "(x), which are represented in Maple by " }{TEXT 263 13 "BesselJ (v, x)" }{TEXT -1 5 " and " }{TEXT 263 12 "BesselY(v,x)" }{TEXT -1 45 ", are solutions of the differential equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "x^2 ;" "6#*$%\"xG\"\"#" }{TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+x; " "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(F+F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+(x^2-v^2)*y = 0;" "6#/,&*&%#dyG\"\"\"%#dxG 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#FY7$F_[m$!+*>$3oBFY7$Fi[m$!+=!['GBFY7$F^\\m$!+geR(4#FY7$Fc\\m$!+&fy#* p\"FY7$Fh\\m$!+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 38 "The general solution of the equation:" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+x;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"* &F%F(*$%\"xGF&F(!\"\"F(F+F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+(x^ 2-1)*y = 0;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&,&*$%\"xG\"\"#F'F'F)F'% \"yGF'F'\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "is " } {XPPEDIT 18 0 "y(x) = C[1]*J[1](x)+C[2]*Y[1](x);" "6#/-%\"yG6#%\"xG,&* &&%\"CG6#\"\"\"F--&%\"JG6#F-6#F'F-F-*&&F+6#\"\"#F--&%\"YG6#F-6#F'F-F- " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 72 "de := x^2*diff(y(x),x$2)+x*diff(y(x),x)+(x^2-1 )*y(x)=0;\ndsolve(de,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/ ,(*&)%\"xG\"\"#\"\"\"-%%diffG6$-%\"yG6#F)-%\"$G6$F)F*F+F+*&F)F+-F-6$F/ F)F+F+*&,&*$F(F+F+F+!\"\"F+F/F+F+\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&%$_C1G\"\"\"-%(BesselJG6$F+F'F+F+*&%$_C2GF+-%( BesselYGF.F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "With the \+ initial condition " }{XPPEDIT 18 0 "y(0) = 0" "6#/-%\"yG6#\"\"!F'" } {TEXT -1 28 " the solution only involves " }{XPPEDIT 18 0 "J[1](x);" " 6#-&%\"JG6#\"\"\"6#%\"xG" }{TEXT -1 41 ", so consider the boundary val ue problem:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT -1 2 " \+ " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+x;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F( *$%\"xGF&F(!\"\"F(F+F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+(x^2-1)* y = 0;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&,&*$%\"xG\"\"#F'F'F)F'%\"yGF' F'\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y(0)=0" "6#/-%\"yG6#\"\"!F' " }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y(8) =1" "6#/-%\"yG6#\"\")\"\"\" " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 122 "de := x^2*diff(y(x),x$2)+x*diff(y(x),x)+(x^2- 1)*y(x)=0;\nbc := y(0)=0,y(8)=1;\ndsolve(\{de,bc\},y(x));\ng := unappl y(rhs(%),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(*&)%\"xG\"\"# \"\"\"-%%diffG6$-%\"yG6#F)-%\"$G6$F)F*F+F+*&F)F+-F-6$F/F)F+F+*&,&*$F(F +F+F+!\"\"F+F/F+F+\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bcG6$/-% \"yG6#\"\"!F*/-F(6#\"\")\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% \"yG6#%\"xG*&-%(BesselJG6$\"\"\"\"\")!\"\"-F*6$F,F'F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&-%(Bess elJG6$\"\"\"\"\")!\"\"-F.6$F09$F0F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "We can find a discrete numerical s olution using " }{TEXT 0 9 "desolveFD" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "soln := desolveFD(\{de,bc\},extrapolate=3,output=points);\ncomparewithfcn(sol n,g(x),x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%solnG777$$\"\"!F(F'7$ $\"+++++S!#5$\"+$o%[a$)F,7$$\"+++++!)F,$\"+'=t>d\"!\"*7$$\"+++++7F4$\" +b_mB@F47$$\"+++++;F4$\"+Jw%)GCF47$$\"+++++?F4$\"++<&zX#F47$$\"+++++CF 4$\"+H])p@#F47$$\"+++++GF4$\"+&pXhu\"F47$$\"+++++KF4$\"+FD#Q6\"F47$$\" +++++OF4$\"+/)f'oSF,7$$F+F4$!+J+r9GF,7$$\"+++++WF4$!+X*=@k)F,7$$\"++++ +[F4$!+63=s7F47$$\"+++++_F4$!+Mqyi9F47$$\"+++++cF4$!+$*y*[U\"F47$$\"++ +++gF4$!+!*G?z6F47$$\"+++++kF4$!+g\\BTxF,7$$\"+++++oF4$!+TNczFF,7$$\"+ ++++sF4$\"+$3)Q:BF,7$$\"+++++wF4$\"+[Vb&y'F,7$$\"\")F($\"\"\"F(" }} {PARA 6 "" 1 "" {TEXT -1 82 " 0 0 function \+ val: 0 rel err: infinity" }}{PARA 6 "" 1 "" {TEXT -1 84 " .4 .8354484683 function val: .8354484742 r el err: 7.0142e-09" }}{PARA 6 "" 1 "" {TEXT -1 84 " .8 1. 571973186 function val: 1.5719731876 rel err: 1.0178e-09" }} {PARA 6 "" 1 "" {TEXT -1 84 " 1.200000000 2.123665255 function \+ val: 2.1236652559 rel err: 4.2380e-10" }}{PARA 6 "" 1 "" {TEXT -1 84 " 1.600000000 2.428847631 function val: 2.4288476313 \+ rel err: 1.2352e-10" }}{PARA 6 "" 1 "" {TEXT -1 84 " 2.000000000 \+ 2.457951700 function val: 2.4579517006 rel err: 2.4411e-10" }}{PARA 6 "" 1 "" {TEXT -1 84 " 2.400000000 2.216985029 functio n val: 2.2169850288 rel err: 9.0213e-11" }}{PARA 6 "" 1 "" {TEXT -1 84 " 2.800000000 1.746145695 function val: 1.7461456 946 rel err: 2.2908e-10" }}{PARA 6 "" 1 "" {TEXT -1 84 " 3.200000 000 1.113822527 function val: 1.113822527 rel err: 0.0000 e-01" }}{PARA 6 "" 1 "" {TEXT -1 84 " 3.600000000 .4068659804 f unction val: .4068659799 rel err: 1.2781e-09" }}{PARA 6 "" 1 " " {TEXT -1 84 " 4.000000000 -.2814710031 function val: -.28147 10036 rel err: 1.7053e-09" }}{PARA 6 "" 1 "" {TEXT -1 84 " 4.4000 00000 -.8642118945 function val: -.8642118949 rel err: 4.39 71e-10" }}{PARA 6 "" 1 "" {TEXT -1 84 " 4.800000000 -1.272180811 \+ function val: -1.2721808113 rel err: 2.3582e-10" }}{PARA 6 "" 1 "" {TEXT -1 84 " 5.200000000 -1.462787034 function val: -1.4627 87034 rel err: 0.0000e-01" }}{PARA 6 "" 1 "" {TEXT -1 84 " 5.600 000000 -1.424897893 function val: -1.4248978932 rel err: 1.4 036e-10" }}{PARA 6 "" 1 "" {TEXT -1 84 " 6.000000000 -1.179202890 \+ function val: -1.1792028892 rel err: 6.7842e-10" }}{PARA 6 "" 1 "" {TEXT -1 84 " 6.400000000 -.774123496 function val: -.77 41234956 rel err: 5.1671e-10" }}{PARA 6 "" 1 "" {TEXT -1 84 " 6.8 00000000 -.2779563541 function val: -.2779563536 rel err: 1 .6549e-09" }}{PARA 6 "" 1 "" {TEXT -1 84 " 7.200000000 .231538808 3 function val: .2315388087 rel err: 1.6412e-09" }}{PARA 6 " " 1 "" {TEXT -1 84 " 7.600000000 .6785554348 function val: . 678555435 rel err: 2.6527e-10" }}{PARA 6 "" 1 "" {TEXT -1 84 " 8 1 function val: 1 rel err: 1.0000e-11" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 6 "" 1 "" {TEXT -1 48 " Maximum relative error: 7.0142e-09" }} {PARA 6 "" 1 "" {TEXT -1 58 " obtained for the input valu e: .4 " }}{PARA 6 "" 1 "" {TEXT -1 67 " exclud ing any cases where the function value is zero." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "The points of the discret e solution appear to lie along the curve for the analytical solution. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "plot([g(x),soln],x=0..8,color=[red,blue],\n styl e=[line,point],symbol=circle);" }}{PARA 13 "" 1 "" {GLPLOT2D 431 312 312 {PLOTDATA 2 "6'-%'CURVESG6%7en7$$\"\"!F)F(7$$\"+MBxV5E$F-$\"+ay7doF-7$$\"+MAKn\\F-$\"+v8?E5!\"*7$$\"+Nc$\\o'F- $\"+eTUY8F:7$$\"+=bQ%R)F-$\"+.W!ej\"F:7$$\"+&Qk#z**F-$\"+o9es=F:7$$\"+ l9.i6F:$\"+#>T63#F:7$$\"+>\"\\F:$\"+[XZg#\\#F:$\"+$>Fj7#F:7$$\"+Fn*Gn#F:$\"+`:* p\">F:7$$\"+2xiDGF:$\"+SBi42VsTF:$!+Q+&>]&F-7$$\"+O&pfK%F:$!+7n'zn(F-7$$\"+kcz\"\\%F:$!+w *pIv*F-7$$\"+\"G5Jm%F:$!+&)=^b6F:7$$\"+6#32$[F:$!+%Qd[H\"F:7$$\"+Ey'G* \\F:$!+43n#R\"F:7$$\"+HJ*G3&F:$!+*fj2V\"F:7$$\"+J%=H<&F:$!+Wa9d9F:7$$ \"+q,\"QD&F:$!+9R&3Z\"F:7$$\"+3>qM`F:$!+\\E:v9F:7$$\"+/62@aF:$!+8lZp9F :7$$\"+,.W2bF:$!+c6R`9F:7$$\"+IOq&e&F:$!+yV/I9F:7$$\"+fp'Rm&F:$!+;qg)R \"F:7$$\"+T>4NeF:$!+[z6.8F:7$$\"+8s5'*fF:$!+J'fC=\"F:7$$\"+mXTkhF:$!+B #\\%G5F:7$$\"+od'*GjF:$!+1cc[&)F-7$$\"+EcB,lF:$!+TwjTlF-7$$\"+v>:nmF:$ !+u(37[%F-7$$\"+0a#o$oF:$!+*y_SI#F-7$$\"+`Q40qF:$!+HF3V8!#67$$\"+\"3:( frF:$\"+.Ij==F-7$$\"+e%GpL(F:$\"+bfM^RF-7$$\"+:-V&\\(F:$\"+0\"Qdr&F-7$ $\"+ZhUkwF:$\"+OH)>S(F-7$$\"+d\"!# <7$$\"3%**************>\"Fi^l$\"39+++b_mB@Fi^l7$$\"33+++++++;Fi^l$\"3, +++Jw%)GCFi^l7$$\"\"#F)$\"3=++++<&zX#Fi^l7$$\"3!**************R#Fi^l$ \"3%*******G])p@#Fi^l7$$\"3#)*************z#Fi^l$\"31+++&pXhu\"Fi^l7$$ \"3;+++++++KFi^l$\"3&******p_AQ6\"Fi^l7$$\"33+++++++OFi^l$\"3%******R! )f'oSFa^l7$$\"\"%F)$!31+++J+r9GFa^l7$$\"3M+++++++WFi^l$!3A+++X*=@k)Fa^ l7$$\"3#)*************z%Fi^l$!3#******4\"3=s7Fi^l7$$\"3;+++++++_Fi^l$! 3++++Mqyi9Fi^l7$$\"3k*************f&Fi^l$!3.+++$*y*[U\"Fi^l7$$\"\"'F)$ !3++++!*G?z6Fi^l7$$\"3M+++++++kFi^l$!3m******f\\BTxFa^l7$$\"3#)******* ******z'Fi^l$!3C+++TNczFFa^l7$$\"3;+++++++sFi^l$\"35+++$3)Q:BFa^l7$$\" 3k*************f(Fi^l$\"3C+++[Vb&y'Fa^lF[]l-Fa]l6&Fc]lF(F(Fd]l-Fh]l6#% &POINTG-%+AXESLABELSG6$Q\"x6\"Q!Fhdl-%'SYMBOLG6#%'CIRCLEG-%%VIEWG6$;F( F\\]l%(DEFAULTG" 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}{PARA 0 "" 0 "" {TEXT -1 49 "The general solution of the differentia l equation" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+x;" "6# ,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(F+F(" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "dy/dx+(x^2-9)*y = 0;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F '*&,&*$%\"xG\"\"#F'\"\"*F)F'%\"yGF'F'\"\"!" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 14 "has the form " }{XPPEDIT 18 0 "y(x) = C[1]*J[3]( x)+C[2]*Y[3](x);" "6#/-%\"yG6#%\"xG,&*&&%\"CG6#\"\"\"F--&%\"JG6#\"\"$6 #F'F-F-*&&F+6#\"\"#F--&%\"YG6#F26#F'F-F-" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "de : = diff(y(x),x$2)+x^2*diff(y(x),x)+x*y(x)=0;\nbc := y(0)=0,y(1)=1;\ndso lve(\{de,bc\},y(x)):\ng := unapply(rhs(%),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(*&)%\"xG\"\"#\"\"\"-%%diffG6$-%\"yG6#F)-%\"$G6 $F)F*F+F+*&F)F+-F-6$F/F)F+F+*&,&*$F(F+F+\"\"*!\"\"F+F/F+F+\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&%$_C1G\"\"\"-%(Besse lJG6$\"\"$F'F+F+*&%$_C2GF+-%(BesselYGF.F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "Consider the two point boundary value problem:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^ 2;" "6#*$%\"xG\"\"#" }{TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+x; " "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(F+F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+(x^2-9)*y = 0;" "6#/,&*&%#dyG\"\"\"%#dxG! \"\"F'*&,&*$%\"xG\"\"#F'\"\"*F)F'%\"yGF'F'\"\"!" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "y(0) = 0" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "y(5) = 1" "6#/-%\"yG6#\"\"&\"\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "de := x^2*diff(y(x),x$2)+x*diff(y(x),x)+(x^2-9)*y(x)=0;\nbc := y( 0)=0,y(5)=1;\ndsolve(\{de,bc\},y(x));\ng := unapply(rhs(%),x);\nevalf( g(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(*&)%\"xG\"\"#\"\"\" -%%diffG6$-%\"yG6#F)-%\"$G6$F)F*F+F+*&F)F+-F-6$F/F)F+F+*&,&*$F(F+F+\" \"*!\"\"F+F/F+F+\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bcG6$/-%\" yG6#\"\"!F*/-F(6#\"\"&\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"y G6#%\"xG,$*(\"#D\"\"\",&*&\"#% \"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,$*(\"#D\"\"\",&*&\"# " 0 "" {MPLTEXT 1 0 84 "soln := desolveFD(\{de,b c\},extrapolate=3,output=points):\ncomparewithfcn(soln,g(x),x);" }} {PARA 6 "" 1 "" {TEXT -1 79 " 0 0 function va l: 0 rel err: infinity" }}{PARA 6 "" 1 "" {TEXT -1 81 " .25 .000888771 function val: .000888771 rel err : 2.7004e-09" }}{PARA 6 "" 1 "" {TEXT -1 81 " .5 .0070271 67 function val: .007027167 rel err: 2.8461e-10" }}{PARA 6 " " 1 "" {TEXT -1 81 " .75 .023255639 function val: .02 3255639 rel err: 4.3000e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " 1 \+ .053623024 function val: .053623024 rel err: 5.594 6e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " 1.25 .101055931 fu nction val: .101055931 rel err: 9.8955e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " 1.5 .167101789 function val: .167101788 rel err: 5.9844e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " 1.75 \+ .251762831 function val: .251762831 rel err: 3.9720e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " 2 .353432598 function \+ val: .353432597 rel err: 2.8294e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " 2.25 .468940123 function val: .468940123 r el err: 4.2649e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " 2.5 .5 93700245 function val: .593700245 rel err: 5.0531e-10" }} {PARA 6 "" 1 "" {TEXT -1 81 " 2.75 .72196175 function va l: .721961749 rel err: 5.5405e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " 3 .847138886 function val: .847138886 rel \+ err: 3.5413e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " 3.25 .9622 06431 function val: .962206431 rel err: 3.1178e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " 3.5 1.060134329 function val: 1 .060134328 rel err: 9.4328e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " 3 .75 1.134335228 function val: 1.134335227 rel err: 8. 8157e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " 4 1.179097176 \+ function val: 1.179097176 rel err: 0.0000e-01" }}{PARA 6 "" 1 " " {TEXT -1 81 " 4.25 1.189974326 function val: 1.1899743 26 rel err: 0.0000e-01" }}{PARA 6 "" 1 "" {TEXT -1 81 " 4.5 \+ 1.164110792 function val: 1.164110792 rel err: 0.0000e-01 " }}{PARA 6 "" 1 "" {TEXT -1 81 " 4.75 1.100476629 functio n val: 1.100476629 rel err: 0.0000e-01" }}{PARA 6 "" 1 "" {TEXT -1 81 " 5 1 function val: 1 r el err: 4.0000e-10" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 6 " " 1 "" {TEXT -1 48 " Maximum relative error: 2.7004e-09" }}{PARA 6 "" 1 "" {TEXT -1 57 " obtained for the input va lue: .25 " }}{PARA 6 "" 1 "" {TEXT -1 67 " exclu ding any cases where the function value is zero." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "The points of the discret e solution appear to lie along the curve for the analytical solution. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "plot([g(x),soln],x=0..5,color=[red,blue],style=[line,point],\n symbol=circle);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 270 270 {PLOTDATA 2 "6'-%'CURVESG6%7S7$$\"\"!F)F(7$$\"+4x&)*3\"!#5$\"+?bv'Q(!# 97$$\"+uq8Q?F-$\"+c>8A[!#87$$\"+(RwX5$F-$\"+5GY)p\"!#77$$\"+sZ3yTF-$\" +b+g>TF<7$$\"+]4\\Y_F-$\"+cen0\")F<7$$\"+U-/PiF-$\"+/u7_8!#67$$\"+fmpi sF-$\"+v(*Q;@FL7$$\"+#*>VB$)F-$\"+ljU6Fin$\"+)oyg%yFL7$$\"+j^Z]7Fin$\"+ Qbj65F-7$$\"+)=h(e8Fin$\"+9fku7F-7$$\"+Q[6j9Fin$\"+]*)oh:F-7$$\"+\\z(y b\"Fin$\"+d^y]=F-7$$\"+b/cq;Fin$\"+[x3HAF-7$$\"+Fin$\"++aFFMF-7$$\"+\"f#=$3#Fin$\"+ 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-BO&Fj\\l7$$\"3+++++++]7!#<$\"3-+++5$f0,\"F[\\l7$$\"3++++++++:Fc]l$\"3 ))*****\\)y,r;F[\\l7$$\"3+++++++]JGw^#F[\\l7$$\"\"#F )$\"3,+++vfKMNF[\\l7$$\"3+++++++]AFc]l$\"3'******4B,%*o%F[\\l7$$Fj[lFc ]l$\"3o*****4X-q$fF[\\l7$$\"3+++++++]FFc]l$\"3G+++&\\<'>sF[\\l7$$\"\"$ F)$\"3!)******f))Qr%)F[\\l7$$\"3+++++++]KFc]l$\"3^*****RJk?i*F[\\l7$$ \"3++++++++NFc]l$\"3/+++HV8g5Fc]l7$$\"3+++++++]PFc]l$\"34+++G_LM6Fc]l7 $$\"\"%F)$\"3'******f<(4z6Fc]l7$$\"3+++++++]UFc]l$\"3'******fKu**=\"Fc ]l7$$\"3++++++++XFc]l$\"3++++#z5T;\"Fc]l7$$\"3+++++++]ZFc]l$\"34+++HmZ +6Fc]l7$FfzF\\]l-F[[l6&F][lF(F(F^[l-Fb[l6#%&POINTG-%+AXESLABELSG6$Q\"x 6\"Q!Febl-%'SYMBOLG6#%'CIRCLEG-%%VIEWG6$;F(Ffz%(DEFAULTG" 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 256 " " 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+x^2;" "6#,&*(%\" dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(*$F+F&F(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx+x*y = 0;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&%\"xGF '%\"yGF'F'\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = 0" "6#/-%\"y G6#\"\"!F'" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y(5) = 1/2;" "6#/-%\"yG6# \"\"&*&\"\"\"F)\"\"#!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "The analytical solution involves a Bessel function of the 2nd kind. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "de := diff(y(x),x$2)+x^2*di ff(y(x),x)+sin(x)*y(x)=0;\nbc := y(0)=0,y(1)=1;\ndsolve(\{de,bc\},y(x) ):\ng := unapply(rhs(%),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/ ,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&)F-F1F2-F(6$F*F-F2F2*& F-F2F*F2F2\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bcG6$/-%\"yG6#\" \"!F*/-F(6#\"\"&#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG f*6#%\"xG6\"6$%)operatorG%&arrowGF(,$*&#\"\"\"\"#5F/*.-%$expG6##!$D\" \"\"'!\"\"\"\"&#F/\"\"#-%(BesselIG6$#F/F7#\"$D\"F7F8-F36#,$*&#F/F7F/*$ )9$\"\"$F/F/F8F/FIF:-F=6$F?,$*&F?F/FGF/F/F/F/F/F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "We can find a discre te numerical solution using " }{TEXT 0 9 "desolveFD" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "soln := desolveFD(\{de,bc\},steps=50,extrapolate=2,output=points ):\ncomparewithfcn(soln,g(x),x);" }}{PARA 6 "" 1 "" {TEXT -1 79 " 0 \+ 0 function val: 0 rel err: infi nity" }}{PARA 6 "" 1 "" {TEXT -1 81 " .1 .181228696 fun ction val: .181228696 rel err: 0.0000e-01" }}{PARA 6 "" 1 "" {TEXT -1 81 " .2 .362034908 function val: .362034907 rel err: 8.2865e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " .3 \+ .541337558 function val: .541337558 rel err: 3.6946e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " .4 .717360486 function \+ val: .717360486 rel err: 1.3940e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " .5 .88769125 function val: .88769125 r el err: 5.6326e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " .6 1.0 49389215 function val: 1.049389215 rel err: 0.0000e-01" }} {PARA 6 "" 1 "" {TEXT -1 81 " .7 1.19915218 function va l: 1.199152179 rel err: 8.3392e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " .8 1.333542796 function val: 1.333542796 rel \+ err: 0.0000e-01" }}{PARA 6 "" 1 "" {TEXT -1 81 " .9 1.4492 65303 function val: 1.449265302 rel err: 6.9000e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " 1 1.543471007 function val: 1 .543471006 rel err: 6.4789e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " 1 .1 1.614059651 function val: 1.61405965 rel err: 6. 1956e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " 1.2 1.659936143 \+ function val: 1.659936142 rel err: 6.0243e-10" }}{PARA 6 "" 1 " " {TEXT -1 81 " 1.3 1.68118114 function val: 1.6811811 38 rel err: 1.1896e-09" }}{PARA 6 "" 1 "" {TEXT -1 81 " 1.4 \+ 1.679101457 function val: 1.679101456 rel err: 5.9556e-10 " }}{PARA 6 "" 1 "" {TEXT -1 81 " 1.5 1.656142545 functio n val: 1.656142543 rel err: 1.2076e-09" }}{PARA 6 "" 1 "" {TEXT -1 81 " 1.6 1.61566783 function val: 1.615667828 r el err: 1.2379e-09" }}{PARA 6 "" 1 "" {TEXT -1 81 " 1.7 1.5 61633963 function val: 1.561633961 rel err: 1.2807e-09" }} {PARA 6 "" 1 "" {TEXT -1 81 " 1.8 1.49821098 function va l: 1.498210978 rel err: 1.3349e-09" }}{PARA 6 "" 1 "" {TEXT -1 81 " 1.9 1.4294065 function val: 1.429406499 rel \+ err: 6.9959e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " 2 1.3587 50385 function val: 1.358750384 rel err: 7.3597e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " 2.1 1.289081109 function val: 1 .289081111 rel err: 1.5515e-09" }}{PARA 6 "" 1 "" {TEXT -1 81 " 2 .2 1.222451841 function val: 1.222451843 rel err: 1. 6361e-09" }}{PARA 6 "" 1 "" {TEXT -1 81 " 2.3 1.160149356 \+ function val: 1.160149358 rel err: 1.7239e-09" }}{PARA 6 "" 1 " " {TEXT -1 81 " 2.4 1.102799015 function val: 1.1027990 16 rel err: 9.0678e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " 2.5 \+ 1.050518659 function val: 1.05051866 rel err: 9.5191e-10 " }}{PARA 6 "" 1 "" {TEXT -1 81 " 2.6 1.0030843 functio n val: 1.0030843 rel err: 0.0000e-01" }}{PARA 6 "" 1 "" {TEXT -1 81 " 2.7 .960078804 function val: .960078803 r el err: 1.0416e-09" }}{PARA 6 "" 1 "" {TEXT -1 81 " 2.8 .9 2100701 function val: .921007009 rel err: 1.5201e-09" }} {PARA 6 "" 1 "" {TEXT -1 81 " 2.9 .885372761 function va l: .88537276 rel err: 1.4683e-09" }}{PARA 6 "" 1 "" {TEXT -1 81 " 3 .852722052 function val: .85272205 rel \+ err: 1.4073e-09" }}{PARA 6 "" 1 "" {TEXT -1 81 " 3.1 .8226 60929 function val: .822660928 rel err: 1.3371e-09" }}{PARA 6 "" 1 "" {TEXT -1 81 " 3.2 .79485751 function val: \+ .794857509 rel err: 1.3839e-09" }}{PARA 6 "" 1 "" {TEXT -1 81 " 3 .3 .769035826 function val: .769035826 rel err: 1. 0403e-09" }}{PARA 6 "" 1 "" {TEXT -1 81 " 3.4 .744966764 \+ function val: .744966763 rel err: 8.0541e-10" }}{PARA 6 "" 1 " " {TEXT -1 81 " 3.5 .722459019 function val: .7224590 19 rel err: 8.3050e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " 3.6 \+ .701351363 function val: .701351362 rel err: 7.1291e-10 " }}{PARA 6 "" 1 "" {TEXT -1 81 " 3.7 .68150648 functio n val: .681506479 rel err: 5.8693e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " 3.8 .662806248 function val: .662806248 r el err: 6.0349e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " 3.9 .6 45148148 function val: .645148148 rel err: 0.0000e-01" }} {PARA 6 "" 1 "" {TEXT -1 81 " 4 .628442506 function va l: .628442505 rel err: 4.7737e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " 4.1 .612610367 function val: .612610366 rel \+ err: 1.3059e-09" }}{PARA 6 "" 1 "" {TEXT -1 81 " 4.2 .5975 8181 function val: .59758181 rel err: 5.0202e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " 4.3 .5832946 function val: \+ .5832946 rel err: 1.3715e-09" }}{PARA 6 "" 1 "" {TEXT -1 81 " 4 .4 .569693093 function val: .569693093 rel err: 1. 2287e-09" }}{PARA 6 "" 1 "" {TEXT -1 81 " 4.5 .556727336 \+ function val: .556727336 rel err: 1.7962e-10" }}{PARA 6 "" 1 " " {TEXT -1 81 " 4.6 .544352318 function val: .5443523 18 rel err: 9.1852e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " 4.7 \+ .532527348 function val: .532527348 rel err: 1.8778e-10 " }}{PARA 6 "" 1 "" {TEXT -1 81 " 4.8 .521215521 functio n val: .521215521 rel err: 5.7558e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " 4.9 .510383274 function val: .510383274 r el err: 3.9186e-10" }}{PARA 6 "" 1 "" {TEXT -1 81 " 5 .5 function val: .5 rel err: 8.0000e-10" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 6 "" 1 "" {TEXT -1 48 " \+ Maximum relative error: 1.7239e-09" }}{PARA 6 "" 1 "" {TEXT -1 57 " obtained for the input value: 2.3 \+ " }}{PARA 6 "" 1 "" {TEXT -1 67 " excluding any cases whe re the function value is zero." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 167 "We can also obtain a numerical solution \+ in the form of a procedure. The graph of the numerical solution appear s to coincide with the graph of the analytical solution. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "gn : = desolveFD(\{de,bc\},output=localtaylor);\nplot(['gn'(x),g(x)],x=0..5 ,color=[red,aquamarine],thickness=[1,2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 413 291 291 {PLOTDATA 2 "6&-%'CURVESG6%7in7$$\"\"!F)F(7$$\"+ X&)G\\a!#6$\"+@U0x)*F-7$$\"+4x&)*3\"!#5$\"+sz.v>F37$$\"+#R(*Rc\"F3$\"+ qx2LGF37$$\"+uq8Q?F3$\"+6s4*o$F37$$\"+NnNrDF3$\"+()GjZYF37$$\"+(RwX5$F 3$\"+%Hc$*f&F37$$\"+%eI8k$F3$\"+=$Gua'F37$$\"+sZ3yTF3$\"+#o()=[(F37$$ \"+hyG7ZF3$\"+)zVB $)F3$\"+PJ;t8F]o7$$\"+Mbw!Q*F3$\"+@c!y[\"F]o7$$\"+0j$o/\"F]o$\"+(4l&z: F]o7$$\"+_>jU6F]o$\"+LImO;F]o7$$\"+eNb'>\"F]o$\"+,)o(e;F]o7$$\"+j^Z]7F ]o$\"+F]o$\"+idE;B\"F]o7$$\"+uI, $H#F]o$\"+&\\XV;\"F]o7$$\"+rSS\"R#F]o$\"+E&Gv5\"F]o7$$\"+`?`(\\#F]o$\" +m\"\\<0\"F]o7$$\"++#pxg#F]o$\"+dd!f***F37$$\"+g4t.FF]o$\"+jb`&e*F37$$ \"+!Hst!GF]o$\"+BWo#=*F37$$\"+ERW9HF]o$\"+kM#[!))F37$$\"+KE>>IF]o$\"+i yin%)F37$$\"+#RU07$F]o$\"+4+vn\")F37$$\"+?S2LKF]o$\"+&G+6'yF37$$\"+$p) 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*3V`F37$Fe]l$\"+3*\\HA&F3Fi]lF^^l-Fd^l6&Ff^l$\")p:#R%Fi^l$\")`B)e)Fi^l $\")fqkdFi^l-F[_l6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q!Fjhl-%%VIEWG6$;F(F_^l %(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 44.000000 0 0 "C urve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 49 "xx := evalf(Pi);\ngn(xx);\nevalf(evalf[13](g (xx)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"+aEfTJ!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+-^R3\")!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+-^R3\")!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "Airy fu nction examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 19 "The Airy functions " }{XPPEDIT 18 0 "Ai(x )" "6#-%#AiG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Bi(x)" "6#-%# BiG6#%\"xG" }{TEXT -1 36 ", which are represented in Maple by " } {TEXT 263 9 "AiryAi(x)" }{TEXT -1 5 " and " }{TEXT 263 9 "AiryBi(x)" } {TEXT -1 45 ", are solutions of the differential equation:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2) = x*y;" "6#/ *(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"*&F+F(F'F(" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "de := diff(y(x),x$2)=x*y(x);\ndsolve(de,y(x));\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$ F,\"\"#*&F,\"\"\"F)F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"x G,&*&%$_C1G\"\"\"-%'AiryAiGF&F+F+*&%$_C2GF+-%'AiryBiGF&F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "Diff(AiryAi(x),x$2)=diff(AiryAi(x), x$2);\nDiff(AiryBi(x),x$2)=diff(AiryBi(x),x$2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%'AiryAiG6#%\"xG-%\"$G6$F*\"\"#*&F*\"\"\"F' F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%'AiryBiG6#%\"xG-%\" $G6$F*\"\"#*&F*\"\"\"F'F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "plot([AiryAi(x),AiryBi(x)],x=-10.. 3,y=-1..2,color=[red,COLOR(RGB,0,.86,0)],\n title=\"Airy functions\", legend=[\"AiryAi(x)\",\"AiryBi(x)\"]);" }}{PARA 13 "" 1 "" {GLPLOT2D 614 453 453 {PLOTDATA 2 "6'-%'CURVESG6%7`t7$$!#5\"\"!$\"+\\Q7CS!#67$$! +\\#f\"H**!\"*$\"+C]F#4\"F)7$$!+)\\=$e)*F1$\"+Rh(zs\"F)7$$!+ZxZ(y*F1$ \"+OgdyAF)7$$!+'*pj;(*F1$\"+$F)7$$!+>DsJ&*F1$\"+k.,5KF)7$$ !+!)HJ;&*F1$\"+lNi/KF)7$$!+SM!4]*F1$\"+!o'*>>$F)7$$!++R\\&[*F1$\"+5*p@ <$F)7$$!+hV3q%*F1$\"+v&*>XJF)7$$!+/ew+%*F1$\"+J'Hy$HF)7$$!+[sWJ$*F1$\" +kxB)f#F)7$$!+#pG@E*F1$\"+V\"eD9#F)7$$!+P,\"G>*F1$\"+f)[=f\"F)7$$!+#4K I7*F1$\"+J\">wm*F-7$$!+[SD`!*F1$\"+Tf`))HF-7$$!+/gZ$)*)F1$!+\"*>=AQF-7 $$!+gzp8*)F1$!++2nY5F)7$$!+e:DW))F1$!+RR5j;F)7$$!+d^![x)F1$!+@L))3AF)7 $$!+b(e`q)F1$!+t-LF)7$$!+U_vU%)F1$!+dT_v KF)7$$!+!Ri0T)F1$!+3a@?KF)7$$!+Q&p$y$)F1$!+Kv&o8$F)7$$!+r=q6$)F1$!+.G* )yGF)7$$!+/U.X#)F1$!+-w)[^#F)7$$!+PlOy\")F1$!+*Q$)*e?F)7$$!+q))p6\")F1 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abs err: 1.1860e-08" }}{PARA 6 "" 1 "" {TEXT -1 84 " 4.900000000 .8466255344 function val: \+ .8466255474 abs err: 1.3020e-08" }}{PARA 6 "" 1 "" {TEXT -1 84 " 5.000000000 .7680271465 function val: .7680271599 abs e rr: 1.3400e-08" }}{PARA 6 "" 1 "" {TEXT -1 84 " 5.100000000 .6512 034675 function val: .6512034806 abs err: 1.3140e-08" }} {PARA 6 "" 1 "" {TEXT -1 84 " 5.200000000 .5013316101 function \+ val: .5013316222 abs err: 1.2140e-08" }}{PARA 6 "" 1 "" {TEXT -1 84 " 5.300000000 .3255305058 function val: .3255305163 \+ abs err: 1.0480e-08" }}{PARA 6 "" 1 "" {TEXT -1 84 " 5.400000000 \+ .1325831843 function val: .1325831925 abs err: 8.2000e-09" }}{PARA 6 "" 1 "" {TEXT -1 84 " 5.500000000 -.0674586040 functio n val: -.0674585985 abs err: 5.5380e-09" }}{PARA 6 "" 1 "" {TEXT -1 84 " 5.600000000 -.2637739905 function val: -.2637739 88 abs err: 2.5300e-09" }}{PARA 6 "" 1 "" {TEXT -1 84 " 5.700000 000 -.4453552239 function val: -.4453552246 abs err: 6.8000 e-10" }}{PARA 6 "" 1 "" {TEXT -1 84 " 5.800000000 -.6016432899 f unction val: -.6016432937 abs err: 3.8400e-09" }}{PARA 6 "" 1 " " {TEXT -1 84 " 5.900000000 -.7231811279 function val: -.72318 11347 abs err: 6.7600e-09" }}{PARA 6 "" 1 "" {TEXT -1 84 " 6.0000 00000 -.8022438552 function val: -.8022438645 abs err: 9.28 00e-09" }}{PARA 6 "" 1 "" {TEXT -1 84 " 6.100000000 -.8334029082 \+ function val: -.8334029194 abs err: 1.1220e-08" }}{PARA 6 "" 1 "" {TEXT -1 84 " 6.200000000 -.8139812874 function val: -.8139 812998 abs err: 1.2440e-08" }}{PARA 6 "" 1 "" {TEXT -1 84 " 6.300 000000 -.7443604715 function val: -.7443604843 abs err: 1.2 840e-08" }}{PARA 6 "" 1 "" {TEXT -1 84 " 6.400000000 -.6281061695 \+ function val: -.6281061819 abs err: 1.2380e-08" }}{PARA 6 "" 1 "" {TEXT -1 84 " 6.500000000 -.4718898024 function val: -.47 18898135 abs err: 1.1080e-08" }}{PARA 6 "" 1 "" {TEXT -1 84 " 6.6 00000000 -.2851950759 function val: -.2851950849 abs err: 9 .0200e-09" }}{PARA 6 "" 1 "" {TEXT -1 84 " 6.700000000 -.079813585 8 function val: -.0798135921 abs err: 6.2700e-09" }}{PARA 6 " " 1 "" {TEXT -1 84 " 6.800000000 .1308507992 function val: . 1308507961 abs err: 3.0900e-09" }}{PARA 6 "" 1 "" {TEXT -1 84 " 6 .900000000 .3326331074 function val: .3326331076 abs err: 2.2000e-10" }}{PARA 6 "" 1 "" {TEXT -1 84 " 7.000000000 .5115635 216 function val: .5115635253 abs err: 3.6700e-09" }}{PARA 6 "" 1 "" {TEXT -1 84 " 7.100000000 .6548659037 function val: \+ .6548659105 abs err: 6.7600e-09" }}{PARA 6 "" 1 "" {TEXT -1 84 " \+ 7.200000000 .7519271181 function val: .7519271275 abs err : 9.3600e-09" }}{PARA 6 "" 1 "" {TEXT -1 84 " 7.300000000 .795161 8737 function val: .795161885 abs err: 1.1260e-08" }}{PARA 6 "" 1 "" {TEXT -1 84 " 7.400000000 .78069965 function val: \+ .7806996623 abs err: 1.2270e-08" }}{PARA 6 "" 1 "" {TEXT -1 84 " 7.500000000 .7088282513 function val: .7088282636 abs e rr: 1.2300e-08" }}{PARA 6 "" 1 "" {TEXT -1 84 " 7.600000000 .5841 42618 function val: .5841426294 abs err: 1.1390e-08" }} {PARA 6 "" 1 "" {TEXT -1 84 " 7.700000000 .4153671721 function \+ val: .4153671816 abs err: 9.5200e-09" }}{PARA 6 "" 1 "" {TEXT -1 84 " 7.800000000 .2148440812 function val: .214844088 \+ abs err: 6.8200e-09" }}{PARA 6 "" 1 "" {TEXT -1 84 " 7.900000000 \+ -.0022932211 function val: -.0022932175 abs err: 3.5730e-09" }}{PARA 6 "" 1 "" {TEXT -1 84 " 8.000000000 -.2192141849 functio n val: -.2192141848 abs err: 8.0000e-11" }}{PARA 6 "" 1 "" {TEXT -1 84 " 8.100000000 -.4186797278 function val: -.4186797 313 abs err: 3.5000e-09" }}{PARA 6 "" 1 "" {TEXT -1 84 " 8.200000 000 -.5844299016 function val: -.5844299084 abs err: 6.7600 e-09" }}{PARA 6 "" 1 "" {TEXT -1 84 " 8.300000000 -.7025596086 f unction val: -.7025596181 abs err: 9.4700e-09" }}{PARA 6 "" 1 " " {TEXT -1 84 " 8.400000000 -.7627641261 function val: -.76276 41374 abs err: 1.1290e-08" }}{PARA 6 "" 1 "" {TEXT -1 84 " 8.5000 00000 -.7593389376 function val: -.7593389496 abs err: 1.20 30e-08" }}{PARA 6 "" 1 "" {TEXT -1 84 " 8.600000000 -.6918317636 \+ function val: -.6918317754 abs err: 1.1800e-08" }}{PARA 6 "" 1 "" {TEXT -1 84 " 8.700000000 -.5652684913 function val: -.5652 685016 abs err: 1.0320e-08" }}{PARA 6 "" 1 "" {TEXT -1 84 " 8.800 000000 -.3899076692 function val: -.3899076771 abs err: 7.9 400e-09" }}{PARA 6 "" 1 "" {TEXT -1 84 " 8.900000000 -.1805181016 \+ function val: -.1805181063 abs err: 4.7400e-09" }}{PARA 6 "" 1 "" {TEXT -1 84 " 9.000000000 .0447823350 function val: .04 47823339 abs err: 1.1520e-09" }}{PARA 6 "" 1 "" {TEXT -1 84 " 9.1 00000000 .2660450594 function val: .2660450621 abs err: 2 .6500e-09" }}{PARA 6 "" 1 "" {TEXT -1 84 " 9.200000000 .463245631 function val: .4632456371 abs err: 6.1100e-09" }}{PARA 6 " " 1 "" {TEXT -1 84 " 9.300000000 .6181238177 function val: . 6181238267 abs err: 9.0100e-09" }}{PARA 6 "" 1 "" {TEXT -1 84 " 9 .400000000 .7159394237 function val: .7159394347 abs err: 1.0980e-08" }}{PARA 6 "" 1 "" {TEXT -1 84 " 9.500000000 .7469714 28 function val: .7469714399 abs err: 1.1860e-08" }}{PARA 6 "" 1 "" {TEXT -1 84 " 9.600000000 .7076018425 function val: \+ .707601854 abs err: 1.1490e-08" }}{PARA 6 "" 1 "" {TEXT -1 84 " \+ 9.700000000 .6008564257 function val: .6008564356 abs err : 9.9200e-09" }}{PARA 6 "" 1 "" {TEXT -1 84 " 9.800000000 .436320 281 function val: .4363202883 abs err: 7.3000e-09" }}{PARA 6 "" 1 "" {TEXT -1 84 " 9.900000000 .2294039632 function val: \+ .229403967 abs err: 3.8200e-09" }}{PARA 6 "" 1 "" {TEXT -1 87 " 10 0 function val: -2.0000000000e-12 ab s err: 2.0000e-12" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 6 " " 1 "" {TEXT -1 48 " Maximum absolute error: 1.7500e-08" }}{PARA 6 "" 1 "" {TEXT -1 58 " obtained for the input va lue: 1.500000000 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 104 "The points of the discrete solution can be plotted tog ether with the curve for the analytical solution. 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "The solution of this type of ODE can be expressed in terms of the " }{HYPERLNK 17 "WhittakerW" 2 "WhittakerW" "" }{TEXT -1 5 " and " }{HYPERLNK 17 "WhittakerM" 2 "Wh ittakerM" "" }{TEXT -1 11 " functions." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "de := diff(y(x),x$2)-2*x *diff(y(x),x)+n*y(x)=0;\ndsolve(de,y(x));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*( F1F2F-F2-F(6$F*F-F2!\"\"*&%\"nGF2F*F2F2\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&**%$_C1G\"\"\"-%$expG6#,$*&\"\"#!\"\"F' F1F+F+F'#F2F1-%+WhittakerMG6%,&*&\"\"%F2%\"nGF+F+#F+F9F+F;*$)F'F1F+F+F +**%$_C2GF+F,F+F'F3-%+WhittakerWGF6F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "Under some circumstances the solution is a polynomial." } }{PARA 0 "" 0 "" {TEXT -1 17 "For example, let " }{XPPEDIT 18 0 "n=8" "6#/%\"nG\"\")" }{TEXT -1 42 " and consider the differential equation: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)- 2*x;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(*&F&F(F+F(F, " }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+8*y = 0;" "6#/,&*&%#dyG\"\"\"% #dxG!\"\"F'*&\"\")F'%\"yGF'F'\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "de := dif f(y(x),x$2)-2*x*diff(y(x),x)+8*y(x)=0;\ndsolve(de,y(x));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"# \"\"\"*(F1F2F-F2-F(6$F*F-F2!\"\"*&\"\")F2F*F2F2\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&%$_C1G\"\"\",(F+F+*&\"\"%F+)F'\"\" #F+!\"\"*&#F.\"\"$F+*$)F'F.F+F+F+F+F+*(%$_C2GF+,(F4F+*&\"#7F+F/F+F1*&F .F+F6F+F+F+-%$IntG6$*&F9!\"#-%$expG6#*$F/F+F+F'F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 42 "We can check by direct substitution that \+ " }{XPPEDIT 18 0 "y = 4*x^4-12*x^2+3;" "6#/%\"yG,(*&\"\"%\"\"\"*$%\"x GF'F(F(*&\"#7F(*$F*\"\"#F(!\"\"\"\"$F(" }{TEXT -1 19 " is a solution o f: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2) -2*x;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(*&F&F(F+F(F ," }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+8*y = 0;" "6#/,&*&%#dyG\"\"\" %#dxG!\"\"F'*&\"\")F'%\"yGF'F'\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "de := diff (y(x),x$2)-2*x*diff(y(x),x)+8*y(x)=0;\nsubs(y(x)=4*x^4-12*x^2+3,de);\n simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-% \"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*(F1F2F-F2-F(6$F*F-F2!\"\"*&\"\")F2F*F 2F2\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,,-%%diffG6$,(\"\"$\"\"\" *&\"#7F*)%\"xG\"\"#F*!\"\"*&\"\"%F*)F.F2F*F*-%\"$G6$F.F/F**(F/F*F.F*-F &6$F(F.F*F0\"#CF**&\"#'*F*F-F*F0*&\"#KF*F3F*F*\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/\"\"!F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "de := diff(y(x),x$2)-2*x*diff(y(x) ,x)+8*y(x)=0;\nbc := y(0)=3,D(y)(0)=0;\ndsolve(\{de,bc\},y(x));\ng := \+ unapply(rhs(%),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diff G6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*(F1F2F-F2-F(6$F*F-F2!\"\"*&\"\") F2F*F2F2\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bcG6$/-%\"yG6#\"\" !\"\"$/--%\"DG6#F(F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\" xG,(\"\"$\"\"\"*&\"#7F*)F'\"\"#F*!\"\"*&\"\"%F*)F'F1F*F*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,(\"\"$ \"\"\"*&\"#7F.)9$\"\"#F.!\"\"*&\"\"%F.)F2F6F.F.F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "This polynomial sati sfies the boundary conditions " }{XPPEDIT 18 0 "y(0)=3" "6#/-%\"yG6#\" \"!\"\"$" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y(1)=-5" "6#/-%\"yG6#\" \"\",$\"\"&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "g(0);\ng(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"&" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "Now we fi nd a discrete numerical solution over the interval from " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = 1 ;" "6#/%\"xG\"\"\"" }{TEXT -1 41 " to satisfy the same boundary condit ions " }{XPPEDIT 18 0 "y(0)=3" "6#/-%\"yG6#\"\"!\"\"$" }{TEXT -1 5 " a nd " }{XPPEDIT 18 0 "y(1)=-5" "6#/-%\"yG6#\"\"\",$\"\"&!\"\"" }{TEXT -1 29 " as the polynomial solution " }{XPPEDIT 18 0 "y(x)=4*x^4-12*x^ 2+3" "6#/-%\"yG6#%\"xG,(*&\"\"%\"\"\"*$F'F*F+F+*&\"#7F+*$F'\"\"#F+!\" \"\"\"$F+" }{TEXT -1 82 ". Values of this discrete solution are then c ompare with the polynomial solution. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "de := diff(y(x),x$2)-2* x*diff(y(x),x)+8*y(x)=0;\nbc := y(0)=3,y(1)=-5;\nsoln := desolveFD(\{d e,bc\},extrapolate=2,output=points):\ncomparewithfcn(soln,g(x),x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G 6$F-\"\"#\"\"\"*(F1F2F-F2-F(6$F*F-F2!\"\"*&\"\")F2F*F2F2\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bcG6$/-%\"yG6#\"\"!\"\"$/-F(6#\"\" \"!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operat orG%&arrowGF(,(*&\"\"%\"\"\")9$F.F/F/*&\"#7F/)F1\"\"#F/!\"\"\"\"$F/F(F (F(" }}{PARA 6 "" 1 "" {TEXT -1 84 " 0 3 fu nction val: 3 rel err: 0.0000e-01" }}{PARA 6 "" 1 "" {TEXT -1 84 " .05 2.970025000 function val: 2.970025 \+ rel err: 0.0000e-01" }}{PARA 6 "" 1 "" {TEXT -1 84 " .1 \+ 2.880400000 function val: 2.8804 rel err: 0.0000 e-01" }}{PARA 6 "" 1 "" {TEXT -1 84 " .15 2.732025000 f unction val: 2.732025 rel err: 0.0000e-01" }}{PARA 6 "" 1 " " {TEXT -1 84 " .2 2.526400000 function val: 2.5264 \+ rel err: 0.0000e-01" }}{PARA 6 "" 1 "" {TEXT -1 84 " .25 \+ 2.265625000 function val: 2.265625 rel err: 0.00 00e-01" }}{PARA 6 "" 1 "" {TEXT -1 84 " .3 1.952400000 \+ function val: 1.9524 rel err: 0.0000e-01" }}{PARA 6 "" 1 "" {TEXT -1 84 " .35 1.590025000 function val: 1.5900 25 rel err: 0.0000e-01" }}{PARA 6 "" 1 "" {TEXT -1 84 " .4 \+ 1.182400000 function val: 1.1824 rel err: 0.0 000e-01" }}{PARA 6 "" 1 "" {TEXT -1 84 " .45 .7340249997 \+ function val: .734025 rel err: 4.0871e-10" }}{PARA 6 "" 1 "" {TEXT -1 84 " .5 .2499999997 function val: .25 rel err: 1.2000e-09" }}{PARA 6 "" 1 "" {TEXT -1 84 " .5 5 -.2639750003 function val: -.263975 rel err: 1 .1365e-09" }}{PARA 6 "" 1 "" {TEXT -1 84 " .6 -.801600000 3 function val: -.8016 rel err: 3.7425e-10" }}{PARA 6 " " 1 "" {TEXT -1 84 " .65 -1.355975000 function val: -1. 355975 rel err: 0.0000e-01" }}{PARA 6 "" 1 "" {TEXT -1 84 " \+ .7 -1.919600000 function val: -1.9196 rel err: 0.0000e-01" }}{PARA 6 "" 1 "" {TEXT -1 84 " .75 -2.4843750 00 function val: -2.484375 rel err: 0.0000e-01" }}{PARA 6 "" 1 "" {TEXT -1 84 " .8 -3.041600000 function val: -3 .0416 rel err: 0.0000e-01" }}{PARA 6 "" 1 "" {TEXT -1 84 " \+ .85 -3.581975000 function val: -3.581975 rel err : 0.0000e-01" }}{PARA 6 "" 1 "" {TEXT -1 84 " .9 -4.095600 000 function val: -4.0956 rel err: 0.0000e-01" }}{PARA 6 "" 1 "" {TEXT -1 84 " .95 -4.571975000 function val: \+ -4.571975 rel err: 0.0000e-01" }}{PARA 6 "" 1 "" {TEXT -1 84 " 1 -5 function val: -5 rel e rr: 0.0000e-01" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 6 "" 1 "" {TEXT -1 48 " Maximum relative error: 1.2000e-09" }} {PARA 6 "" 1 "" {TEXT -1 58 " obtained for the input valu e: .5 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 121 "plot([g(x),soln],x=0..1,color=[red,blue],st yle=[line,point],\n symbol =circle);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%' CURVESG6%7S7$$\"\"!F)$\"\"$F)7$$\"3emmm;arz@!#>$\"3q,My6&*H%*H!#<7$$\" 3[LL$e9ui2%F/$\"3Ce$y0$=2!)HF27$$\"3nmmm\"z_\"4iF/$\"39`z1h^z`HF27$$\" 3[mmmT&phN)F/$\"3/R[6TVS;HF27$$\"3CLLe*=)H\\5!#=$\"3Ux)H5qh$oGF27$$\"3 gmm\"z/3uC\"FE$\"3,ntCn_C9GF27$$\"3%)***\\7LRDX\"FE$\"3oDiOghf[FF27$$ \"3]mm\"zR'ok;FE$\"3??!433I0n#F27$$\"3w***\\i5`h(=FE$\"3q.jc#e#F27$ $\"3WLLL3En$4#FE$\"3g!HCs5q;[#F27$$\"3qmm;/RE&G#FE$\"3!z2%G\"yM!z,?F27$$\"3jmm\"z*ev:JFE$\"3YU P]@aus=F27$$\"3?LLL347TLFE$\"3=1(Q3:v-r\"F27$$\"3,LLLLY.KNFE$\"3[t&QIp ?_c\"F27$$\"3w***\\7o7Tv$FE$\"3ytxU)pX#)Q\"F27$$\"3'GLLLQ*o]RFE$\"3j!> Ukg*[C7F27$$\"3A++D\"=lj;%FE$\"3c`EUAl\\P5F27$$\"31++vV&RY2aFE$!3Wd(HH97(o;FE7$$\"39mm;zXu9cFE$!3sg?fpb. bQFE7$$\"3l******\\y))GeFE$!3]4y@MSm`hFE7$$\"3'*)***\\i_QQgFE$!3]%**)[ YzdO%)FE7$$\"3@***\\7y%3TiFE$!3MZ@;5-En5F27$$\"35****\\P![hY'FE$!3w>D^ 3A1=8F27$$\"3kKLL$Qx$omFE$!3U.uvbg8X:F27$$\"3!)*****\\P+V)oFE$!3cb*pk) *p()y\"F27$$\"3?mm\"zpe*zqFE$!3E^UN\"))e+,#F27$$\"3%)*****\\#\\'QH(FE$ !3#f\">l^+%>D#F27$$\"3GKLe9S8&\\(FE$!3g.9i7/!*yCF27$$\"3R***\\i?=bq(FE $!3<#fY\\`W[r#F27$$\"3\"HLL$3s?6zFE$!3zCB-B4gVHF27$$\"3a***\\7`Wl7)FE$ !3)*GB*\\7O.=$F27$$\"3#pmmm'*RRL)FE$!3nDOhRD(\\S$F27$$\"3Qmm;a<.Y&)FE$ !3$oE\"4)\\M0j$F27$$\"3=LLe9tOc()FE$!3!GX%fWsJ\\QF27$$\"3u******\\Qk\\ *)FE$!3mGPL1$z`/%F27$$\"3CLL$3dg6<*FE$!3'GM/e(HTjUF27$$\"3ImmmmxGp$*FE $!3K.7v,bk^WF27$$\"3A++D\"oK0e*FE$!3b/)f[K![WYF27$$\"3A++v=5s#y*FE$!38 `)*=H[n?[F27$$\"\"\"F)$!\"&F)-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%&ST YLEG6#%%LINEG-F$6%77F'7$$\"3G+++++++]F/$\"39++++]-qHF27$$\"3/+++++++5F E$\"3%)**********R!)GF27$$\"3%**************\\\"FE$\"3:++++]-KFF27$$\" 35+++++++?FE$\"3?+++++SEDF27$$\"3++++++++DFE$\"3+++++]ilAF27$$\"3))*** ***********HFE$\"3#***********R_>F27$$\"3w*************\\$FE$\"3-++++] -!f\"F27$$\"3A+++++++SFE$\"3))**********R#=\"F27$$\"35+++++++XFE$\"39+ ++(**\\-M(FE7$$\"3++++++++]FE$\"3-+++(******\\#FE7$$\"3U+++++++bFE$!3% )*****H+](REFE7$$\"3w**************fFE$!3/+++.++;!)FE7$$\"3A+++++++lFE $!3#*********\\(fN\"F27$$\"3a**************pFE$!3(***********f>>F27$$ \"3++++++++vFE$!3+++++]P%[#F27$$\"3U+++++++!)FE$!3')**********fTIF27$$ \"3w*************\\)FE$!3!*********\\(>e$F27$$\"3A+++++++!*FE$!37+++++ g&4%F27$$\"3a*************\\*FE$!37++++](>d%F2Fdz-Fjz6&F\\[lF(F(F][l-F a[l6#%&POINTG-%'SYMBOLG6#%'CIRCLEG-%+AXESLABELSG6$Q\"x6\"Q!Fcbl-%%VIEW G6$;F(Fez%(DEFAULTG" 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 78 "We can also obtain a continuous solution \+ in the form of a numerical procedure." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "de := diff(y(x),x$2)-2* x*diff(y(x),x)+8*y(x)=0;\nbc := y(0)=3,y(1)=-5;\ngn := desolveFD(\{de, bc\},output=localtaylor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,( -%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*(F1F2F-F2-F(6$F*F-F2!\"\" *&\"\")F2F*F2F2\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bcG6$/-%\"y G6#\"\"!\"\"$/-F(6#\"\"\"!\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 116 "The zero of the numerical procedure agre es (to 10 digits) with one of the zeros of the polynomial, as calculat ed by " }{TEXT 0 6 "fsolve" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "fsolve('gn'(x),x=0 .5);\nfsolve(g(x),x=0.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+LiZY_ !#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$!+C,o];!\"*$!+LiZY_!#5$\"+LiZY _F($\"+C,o];F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 180 "With a wider interval whose end points are used to provi de the boundary values we can get a numerical solution to the differen tial equation which has all 4 roots of the polynomial." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "de := \+ diff(y(x),x$2)-2*x*diff(y(x),x)+8*y(x)=0;\nbc := y(-2)=19,y(2)=19;\ngn := desolveFD(\{de,bc\},output=localtaylor);\nplot('gn'(x),x=-2..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"xG-%\" $G6$F-\"\"#\"\"\"*(F1F2F-F2-F(6$F*F-F2!\"\"*&\"\")F2F*F2F2\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bcG6$/-%\"yG6#!\"#\"#>/-F(6#\"\"#F+ " }}{PARA 13 "" 1 "" {GLPLOT2D 360 270 270 {PLOTDATA 2 "6%-%'CURVESG6$ 7io7$$!\"#\"\"!$\"#>F*7$$!+YG?y>!\"*$\"+#p!eHF0$\"+ed% pc\"F37$$!+Q&3Y$>F0$\"+3'**=T\"F37$$!+$Q6G\">F0$\"+;)[UE\"F37$$!+3-)[( =F0$\"+$RtV-\"F37$$!+M!\\p$=F0$\"+KyA`!)F07$$!+h9H%z\"F0$\"+*Grk#eF07$ $!+))Qj^FPQF07$$!+`Np3'***F0$!+Fs&p*\\F07$$!*] 5*H\"*F0$!+emF[o7$$!*G\\N)\\F0$\"+oWMkEF[o7$$ !*ZUs>%F0$\"+g*>,,\"F07$$!*GRXL$F0$\"+`f::F0$\"+BK&***HF07$$\"+]+07UF\\w$\"+^HsyHF07$$\")f`@')F0$\"+ YR-6HF07$$\"*nZ)H;F0$\"+!=aSo#F07$$\"*Ky*eCF0$\"+*\\N!*G#F07$$\"*S^bJ$ F0$\"+W;>H$)F0$!+#39)*Q$F07$$\"*qfa<*F0$!+3\"ovE%F07$$\"*1O0)**F0$!+0NQ %)\\F07$$\"+#G2A3\"F0$!+n,\\nbF07$$\"+#3XL7\"F0$!+4:CtdF07$$\"+$)G[k6F 0$!+#[zq\"fF07$$\"+[.b27F0$!+HO+$*fF07$$\"+7yh]7F0$!+(*Hf$)fF07$$\"++p 4#H\"F0$!+u%f])eF07$$\"+()fdL8F0$!+r#=**o&F07$$\"+WV*fP\"F0$!+u-2\"Q&F 07$$\"+-FT=9F0$!+-d&=&\\F07$$\"+Epa-:F0$!+)[#)Qq$F07$$\"+LA?T:F0$!+p$p `$HF07$$\"+Sv&)z:F0$!+6cLK?F07$$\"+%)3;C;F0$!+GJZ2#)F[o7$$\"+GUYo;F0$ \"+4B\"H#fF[o7$$\"+o'*33V_=F0$\"+)GIG#*) F07$$\"+S2ls=F0$\"+!)=$4,\"F37$$\"+u&pG*=F0$\"+')4[N6F37$$\"+2%)38>F0$ \"+P(zgE\"F37$$\"+1j\"[$>F0$\"+vAM89F37$$\"+/Uac>F0$\"+s]&zc\"F37$$\"+ -@Fy>F0$\"+L(4,t\"F37$$\"\"#F*F+-%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*F\\dl -%+AXESLABELSG6$Q\"x6\"Q!Fadl-%%VIEWG6$;F(Fccl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "fsolve( 'gn'(x),x=-2..-1);\nfsolve('gn'(x),x=-1..0);\nfsolve('gn'(x),x=0..1); \nfsolve('gn'(x),x=1..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+C,o];! \"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+LiZY_!#5" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+LiZY_!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+C ,o];!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "The Legend re equation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 35 "The Legendre equation has the form:" }}{PARA 256 "" 0 "" {TEXT -1 2 "( " }{XPPEDIT 18 0 "x^2-1;" "6#,&*$%\"xG\"\"#\"\" \"F'!\"\"" }{TEXT -1 2 ") " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+2*x;" "6#,&* (%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(*&F&F(F+F(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx-n*(n+1)*y = 0;" "6#/,&*&%#dyG\"\"\"%#dxG! \"\"F'*(%\"nGF',&F+F'F'F'F'%\"yGF'F)\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "The solution of t his type of ODE can be expressed in terms of the " }{HYPERLNK 17 "Lege ndre" 2 "Legendre" "" }{TEXT -1 11 " functions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "de := (x^2-1 )*diff(y(x),x$2)+2*x*diff(y(x),x)-n*(n+1)*y(x)=0;\ndsolve(de,y(x));\n " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(*&,&*$)%\"xG\"\"#\"\"\"F- F-!\"\"F--%%diffG6$-%\"yG6#F+-%\"$G6$F+F,F-F-*(F,F-F+F--F06$F2F+F-F-*( %\"nGF-,&F " 0 "" {MPLTEXT 1 0 18 "ortho poly[P](8,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&#\"%Nk\"$G\"\"\" \"*$)%\"xG\"\")F(F(F(*&#\"%.I\"#KF(*$)F+\"\"'F(F(!\"\"*&#\"%lM\"#kF(*$ )F+\"\"%F(F(F(*&#\"$:$F0F(*$)F+\"\"#F(F(F4#\"#NF'F(" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "We can arrange to get t he polynomial " }{XPPEDIT 18 0 "P(8,x)" "6#-%\"PG6$\"\")%\"xG" }{TEXT -1 43 " as a solution of the differential equation" }}{PARA 256 "" 0 " " {TEXT -1 2 "( " }{XPPEDIT 18 0 "x^2-1;" "6#,&*$%\"xG\"\"#\"\"\"F'!\" \"" }{TEXT -1 2 ") " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+2*x;" "6#,&*(%\"dG \"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(*&F&F(F+F(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx-72*y = 0;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&\"#s F'%\"yGF'F)\"\"!" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 43 "by c hoosing appropriate initial conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "de := (x^2-1)*diff(y(x ),x$2)+2*x*diff(y(x),x)-72*y(x)=0;\nbc := y(0)=35/128,D(y)(0)=0;\ndsol ve(\{de,bc\},y(x)):\ng := unapply(rhs(%),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(*&,&*$)%\"xG\"\"#\"\"\"F-F-!\"\"F--%%diffG6$-% \"yG6#F+-%\"$G6$F+F,F-F-*(F,F-F+F--F06$F2F+F-F-*&\"#sF-F2F-F.\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bcG6$/-%\"yG6#\"\"!#\"#N\"$G\"/--% \"DG6#F(F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%) operatorG%&arrowGF(,,*&#\"%Nk\"$G\"\"\"\"*$)9$\"\")F1F1F1*&#\"%.I\"#KF 1*$)F4\"\"'F1F1!\"\"*&#\"%lM\"#kF1*$)F4\"\"%F1F1F1*&#\"$:$F9F1*$)F4\" \"#F1F1F=#\"#NF0F1F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 40 "Note that the value of the solution for " } {XPPEDIT 18 0 "x = 1/2;" "6#/%\"xG*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 4 " \+ is " }{XPPEDIT 18 0 "y(1/2) = -2413/32768;" "6#/-%\"yG6#*&\"\"\"F(\"\" #!\"\",$*&\"%8CF(\"&oF$F*F*" }{TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "g(1/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!%8C \"&oF$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "Now we find a discrete numerical solution over the interval from \+ " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x=1/2" "6#/%\"xG*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 158 " to satisfy the same boundary conditions as the previous polynomial solution. Val ues of this discrete solution are then compare with the polynomial sol ution. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 188 "de := (x^2-1)*diff(y(x),x$2)+2*x*diff(y(x),x)-72*y(x )=0;\nbc := y(0)=35/128,y(1/2)=-2413/32768;\nsoln := desolveFD(\{de,bc \},steps=20,extrapolate=2,output=points):\ncomparewithfcn(soln,g(x),x) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(*&,&*$)%\"xG\"\"#\"\"\"F -F-!\"\"F--%%diffG6$-%\"yG6#F+-%\"$G6$F+F,F-F-*(F,F-F+F--F06$F2F+F-F-* &\"#sF-F2F-F.\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bcG6$/-%\"yG6 #\"\"!#\"#N\"$G\"/-F(6##\"\"\"\"\"##!%8C\"&oF$" }}{PARA 6 "" 1 "" {TEXT -1 84 " 0 .2734375 function val: .2734375 rel err: 0.0000e-01" }}{PARA 6 "" 1 "" {TEXT -1 84 " .025 \+ .2673062825 function val: .267306282 rel err: 1.7583 e-09" }}{PARA 6 "" 1 "" {TEXT -1 84 " .05 .2491650404 f unction val: .2491650396 rel err: 3.3713e-09" }}{PARA 6 "" 1 " " {TEXT -1 84 " .075 .2197627988 function val: .21976 27976 rel err: 5.3694e-09" }}{PARA 6 "" 1 "" {TEXT -1 84 " .1 \+ .1803207228 function val: .1803207215 rel err: 7.32 03e-09" }}{PARA 6 "" 1 "" {TEXT -1 84 " .125 .1324918447 \+ function val: .1324918433 rel err: 1.0491e-08" }}{PARA 6 "" 1 "" {TEXT -1 84 " .15 .0783057634 function val: .0783 057620 rel err: 1.7879e-08" }}{PARA 6 "" 1 "" {TEXT -1 84 " .175 .0200993999 function val: .0200993987 rel err: 6.2 191e-08" }}{PARA 6 "" 1 "" {TEXT -1 84 " .2 -.0395647990 \+ function val: -.0395648 rel err: 2.5022e-08" }}{PARA 6 "" 1 "" {TEXT -1 84 " .225 -.0979915093 function val: -.09 79915099 rel err: 6.6332e-09" }}{PARA 6 "" 1 "" {TEXT -1 84 " .2 5 -.1524540184 function val: -.1524540186 rel err: 1 .2463e-09" }}{PARA 6 "" 1 "" {TEXT -1 84 " .275 -.200302235 1 function val: -.2003022349 rel err: 1.2481e-09" }}{PARA 6 " " 1 "" {TEXT -1 84 " .3 -.2390745917 function val: -. 239074591 rel err: 2.8443e-09" }}{PARA 6 "" 1 "" {TEXT -1 84 " \+ .325 -.2666108995 function val: -.2666108985 rel err: 3.8633e-09" }}{PARA 6 "" 1 "" {TEXT -1 84 " .35 -.2811629 105 function val: -.2811629092 rel err: 4.7304e-09" }}{PARA 6 "" 1 "" {TEXT -1 84 " .375 -.2814990284 function val: - .2814990268 rel err: 5.6483e-09" }}{PARA 6 "" 1 "" {TEXT -1 84 " \+ .4 -.2669993016 function val: -.2669993 rel err : 5.9925e-09" }}{PARA 6 "" 1 "" {TEXT -1 84 " .425 -.237736 5248 function val: -.2377365234 rel err: 5.8889e-09" }}{PARA 6 "" 1 "" {TEXT -1 84 " .45 -.1945389621 function val: \+ -.1945389609 rel err: 6.1684e-09" }}{PARA 6 "" 1 "" {TEXT -1 84 " .475 -.1390298983 function val: -.1390298976 rel e rr: 5.0349e-09" }}{PARA 6 "" 1 "" {TEXT -1 84 " .5 -.0736 389160 function val: -.0736389161 rel err: 1.0864e-09" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 6 "" 1 "" {TEXT -1 48 " \+ Maximum relative error: 6.2191e-08" }}{PARA 6 "" 1 "" {TEXT -1 58 " obtained for the input value: .175 \+ " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "plot([g(x),soln],x=0..0.5,color=[red,blue],style=[li ne,point],\n symbol=circle);" }}{PARA 13 "" 1 "" {GLPLOT2D 338 175 175 {PLOTDATA 2 "6'-%'CURVESG6%7S7$$\"\"!F)$\"3 +++++]PMF!#=7$$\"3ILLL3x&)*3\"!#>$\"347YuI.pAFF,7$$\"3umm\"H2P\"Q?F0$ \"3'eHT?RxNp#F,7$$\"3MLL$eRwX5$F0$\"33ZCWs\"***REF,7$$\"3CLL$3x%3yTF0$ \"39#=WTX$=kDF,7$$\"3=mm\"z%4\\Y_F0$\"3A]3!=q,vY#F,7$$\"3)HL$eR-/PiF0$ \"3C!op')G%efBF,7$$\"3A***\\il'pisF0$\"3`oWAV[2IAF,7$$\"3`KLe*)>VB$)F0 $\"3m='32Xz!y?F,7$$\"3!))**\\7`l2Q*F0$\"3=v?:IgU4>F,7$$\"3smm;/j$o/\"F ,$\"3#)f9C%fA%>jU6F,$\"34s(y\"\\@QR:F,7$$\"3!*****\\i^Z ]7F,$\"3QBY3!)=%RK\"F,7$$\"3'*****\\(=h(e8F,$\"3==5Phao&4\"F,7$$\"3)** ***\\P[6j9F,$\"3iw_vZwBh')F07$$\"3KL$e*[z(yb\"F,$\"34,)f0!ot4lF07$$\"3 gmm;a/cq;F,$\"3MvH)fJ`y)QF07$$\"3]mmm;t,mF,$!3%\\s-t->\"pLF07$$\"35 +]i!f#=$3#F,$!3kWr2._tFfF07$$\"3/+](=xpe=#F,$!3F$Hw4/UtK)F07$$\"3smm\" H28IH#F,$!3qQp0()e>x5F,7$$\"3km;zpSS\"R#F,$!3=0Ff7Nc%H\"F,7$$\"3GLL3_? `(\\#F,$!3bU_s&eW%>:F,7$$\"3#HLe*)>pxg#F,$!3F1&4^F4/u\"F,7$$\"3u**\\Pf 4t.FF,$!3a4T\")\\F,7$$\"32LLe*Gst!GF,$!3=$yZd$R'35#F,7$$\"3#)***** \\#RW9HF,$!37O,Eq:ppAF,7$$\"3[***\\7j#>>IF,$!39g%Q:(y6;CF,7$$\"3h**\\i !RU07$F,$!3?(\\'f]?tQDF,7$$\"3b***\\(=S2LKF,$!3'f.Rv2$R^EF,7$$\"3Kmmm \"p)=MLF,$!3,$GxAHV.t#F,7$$\"3!*****\\(=]@W$F,$!3kUPE1.C!z#F,7$$\"35L$ e*[$z*RNF,$!3MU$QoMa>#GF,7$$\"3#*****\\iC$pk$F,$!35_V1*et8$GF,7$$\"39m ;H2qcZPF,$!3/c)>2unc\"GF,7$$\"3q**\\7.\"fF&QF,$!3W['e9'=ZtFF,7$$\"3Ymm ;/OgbRF,$!3+D&GEeRmq#F,7$$\"3y**\\ilAFjSF,$!3_sSM=Ip4EF,7$$\"3YLLL$)*p p;%F,$!3#Qir$ogd!\\#F,7$$\"3?LL3xe,tUF,$!3w'>$=b9BVBF,7$$\"3em;HdO=yVF ,$!3k5CZ(3]C<#F,7$$\"3))*****\\#>#[Z%F,$!3g(G;z`.[*>F,7$$\"3immT&G!e&e %F,$!3]=!*=.K:E253Y:F,7$$\"37+]iSjE!z%F,$ !3=(fsK%\\!4H\"F,7$$\"35+]P40O\"*[F,$!3&*oSn0l!3.\"F,7$$\"3++++++++]F, $!3++Dc,;*QO(F0-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%&STYLEG6#%%LINEG- F$6%77F'7$$\"39+++++++DF0$\"3#******\\#G1tEF,7$$\"3G+++++++]F0$\"35+++ //l\"\\#F,7$$\"3s*************\\(F0$\"3#******z)zi(>#F,7$$\"3/+++++++5 F,$\"3++++Gs?.=F,7$$\"3+++++++]7F,$\"3++++Z%=\\K\"F,7$$\"3%*********** ***\\\"F,$\"3L+++PjdIyF07$$\"3))************\\F,7$$\"3y************\\Z F,$!35+++$)*)H!R\"F,7$Fdz$!3/+++-;*QO(F0-Fiz6&F[[lF(F(F\\[l-F`[l6#%&PO INTG-%+AXESLABELSG6$Q\"x6\"Q!Fabl-%'SYMBOLG6#%'CIRCLEG-%%VIEWG6$;F($\" \"&!\"\"%(DEFAULTG" 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 41 "We can also obtain a continuous solution. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "de := (x^2-1)*diff(y(x),x$2)+2*x*diff(y(x),x)-72*y(x)=0;\nbc \+ := y(0)=35/128,y(1/2)=-2413/32768;\ngn := desolveFD(\{de,bc\},output=l ocaltaylor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(*&,&*$)%\"xG \"\"#\"\"\"F-F-!\"\"F--%%diffG6$-%\"yG6#F+-%\"$G6$F+F,F-F-*(F,F-F+F--F 06$F2F+F-F-*&\"#sF-F2F-F.\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b cG6$/-%\"yG6#\"\"!#\"#N\"$G\"/-F(6##\"\"\"\"\"##!%8C\"&oF$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "fs olve('gn'(x),x=0.2);\nfsolve(g(x),x=0.1..0.3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+DkMM=!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+DkM M=!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "With a wider interval we can get a numerical solution to the diff erential equation which has all the roots of the polynomial." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 206 "x x := 99/100: yy := orthopoly[P](8,xx):\nde := (x^2-1)*diff(y(x),x$2)+2 *x*diff(y(x),x)-72*y(x)=0;\nbc := y(-xx)=yy,y(xx)=yy;\nfn := desolveFD (\{de,bc\},extrapolate=3,output=localtaylor);\nplot('fn'(x),x=-xx..xx) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(*&,&*$)%\"xG\"\"#\"\"\"F -F-!\"\"F--%%diffG6$-%\"yG6#F+-%\"$G6$F+F,F-F-*(F,F-F+F--F06$F2F+F-F-* &\"#sF-F2F-F.\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bcG6$/-%\"yG6 ##!#**\"$+\"#\"3(o2xn1Qhr\"\"3+++++++gD/-F(6##\"#**F,F-" }}{PARA 13 " " 1 "" {GLPLOT2D 360 270 270 {PLOTDATA 2 "6%-%'CURVESG6$7]u7$$!+++++** !#5$\"+BVm.nF*7$$!+Ag-t)*F*$\"+mLq8fF*7$$!+W?0Y)*F*$\"+`*yM;&F*7$$!+m! 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