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Geddes\nSymbolic Computation Group\nDepartment of Computer S cience\nUniversity of Waterloo\nWaterloo ON N2L 3G1\nCANADA\nhttp:// daisy.uwaterloo.ca/~kogeddes" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 8 "A bstract" }}{PARA 0 "" 0 "" {TEXT -1 828 "The Maple scientific computat ion environment supports both symbolic and numeric mathematical comput ation. We first discuss the general concept of \"closed form\" soluti ons. Then we present a survey of techniques applied in Maple for solv ing ordinary differential equations (ODEs) in both the symbolic and nu meric paradigms. For the symbolic solution of ODEs, algorithmic metho ds are applied when possible. In addition, large classes of ODEs of i nterest in physical applications are solved by employing classificatio n techniques which have been developed specifically for computer algeb ra systems. For the numeric solution of ODEs, various well-known nume rical methods are available as well as some recently-developed techniq ues which exploit the symbolic capabilities of Maple to achieve a more powerful numerical ODE solver." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 33 "What is a \"Closed Form\" Solution?" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 35 "Example 1: Polynomial Root Finding" }}{PARA 0 "" 0 "" {TEXT -1 65 "The following polynomial is cubic in x, so there are thre e roots." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "cubic_poly := x^3 + (a+2)*x^ 2 + 4*a*(x+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+cubic_polyG,(*$)% \"xG\"\"$\"\"\"\"\"\"*&,&%\"aGF+\"\"#F+F+)F(F/F*F+*&F.F+,&F(F+F+F+F+\" \"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "solve(cubic_poly, x) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%!\"#,&%\"aG#!\"\"\"\"#*$-%%sqrtG6 #,&*$)F%F(\"\"\"\"\"\"F%!\")F0#F1F(,&F%F&F)F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "The result is presented in the form of three explicit \+ solutions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "However, it is sometimes more useful for further computation i f we let solutions which are not rational numbers be expressed " } {TEXT 279 10 "implicitly" }{TEXT -1 21 " in terms of Maple's " }{TEXT 280 6 "RootOf" }{TEXT -1 10 " notation." }}{PARA 0 "" 0 "" {TEXT -1 50 "We can request this mode by setting the following " }{TEXT 281 20 "environment variable" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "_EnvExplicit := false;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-_EnvExplicitG%&falseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "solve(cubic_poly, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$!\"#- %'RootOfG6#,(*$)%#_ZG\"\"#\"\"\"\"\"\"*&%\"aGF-F*F-F-F/F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "It is pe rhaps more readily agreed that the implicit notation is convenient if \+ we consider the case of a quartic polynomial." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "quartic_pol y := x^4 - 1/3*x^3 + 1/2*x^2 + 10*x - 25;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-quartic_polyG,,*$)%\"xG\"\"%\"\"\"\"\"\"*$)F(\"\"$F* #!\"\"F.*$)F(\"\"#F*#F+F3F(\"#5!#DF+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "solve(quartic_poly, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RootOfG6#,,*$)%#_ZG\"\"%\"\"\"\"\"'*$)F)\"\"$F+!\"#*$)F)\"\"# F+F/F)\"#g!$]\"\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "_E nvExplicit := true;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-_EnvExplicit G%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "solve(quartic_p oly, x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6&,(#\"\"\"\"#7F%*$-%%sqrtG6 #*&,(*$),&\"&hT\"F%*$-F)6#\")mRd<\"\"\"\"#5#F%\"\"$F5!#6*$)F/#\"\"#F8F 5\"\"'!%apF%F5*$)F/#\"\"\"F8F5!\"\"F5F$*&%\"IGF%-F)6#*&,**&F.F5F(F5\"# A*&F(F5F;F5F>F'F?F-\"%aVF5*&)F/#\"\"\"F8F5-F)6#F+F5FDF5F$,(F$F%F'F$FE# !\"\"F&,(F$F%F'FV*$-F)6#*&,*FK!#AFM!\"'F'\"%apF-FNF5*&)F/#\"\"\"F8F5-F )6#F+F5FDF5F$,(F$F%F'FVFYFV" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 83 "Further manipulation with the complicate d expressions appearing in the solution of " }{TEXT 282 12 "quartic_po ly" }{TEXT -1 58 " are often difficult. In contrast, manipulation wit h the " }{TEXT 283 6 "RootOf" }{TEXT -1 33 " notation can be straightf orward." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "As one example of the use of implicit notation, consider the follo wing integral." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 28 "r := Int(1/quartic_poly, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG-%$IntG6$*&\"\"\"F),,*$)%\"xG\"\"%F)\"\"\"*$)F -\"\"$F)#!\"\"F2*$)F-\"\"#F)#F/F7F-\"#5!#DF/!\"\"F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "va lue(r);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$-%$sumG6$*&%#_RG\"\"\"-%# lnG6#,,%\"xGF)*$)F(\"\"$\"\"\"#!3DW3$*HF6U\")\",eEr0C%*$)F(\"\"#F2#\"1 DIq&HL%))\\F5F(#!0fQv_?RJ\"\"-k7qX#R$#\"-bEVZ+W\".#z.rt<5F)F)/F(-%'Roo tOfG6#,**$)%#_ZG\"\"%F2\",+#3(\\u%*$)FHF8F2!'`*\\'FH!%aV!\"*F)\"\"'" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 154 "The above result is much more c ompact than an explicit result, and it is also much easier to evaluate and otherwise to manipulate in further computations." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 221 "Of cours e, it we try to solve a polynomial of degree 5 which is irreducible ov er the rationals then the solution cannot be expressed in terms of rad icals, hence implicit notation is necessary regardless of the setting \+ of " }{TEXT 284 12 "_EnvExplicit" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "q := x^5 + \+ x^3 + 2*x^2 + 2*x + 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qG,,*$)% \"xG\"\"&\"\"\"\"\"\"*$)F(\"\"$F*F+*$)F(\"\"#F*F1F(F1F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "solve(q, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RootOfG6#,,*$)%#_ZG\"\"&\"\"\"\"\"\"*$)F)\"\"$F+F,*$ )F)\"\"#F+F2F)F2F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "Alternatively, the polynomial can be solved in approximate floating-point arithmetic by applying the " }{XPPEDIT 18 0 "fsolve" "6#%'fsolveG" }{TEXT -1 62 " command. Here we ask for all \+ the roots in the complex field." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "fsolve(q, x, complex);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6'$!+a1wWt!#5,&$!+bo(48%F%\"\"\"%\"IG$!+ ;b(em'F%,&F'F)F*$\"+;b(em'F%,&$\"+#=dL!yF%F)F*$!+$*4(oE\"!\"*,&F1F)F*$ \"+$*4(oE\"F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 44 "Example 2: Solving Non-polynomia l Equations" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "r := solve(x=cos(x), x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG-%'RootOfG6#,&%#_ZG\"\"\"-%$cosG 6#F)!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "To compute a floating-point approximation for " } {XPPEDIT 18 0 "r;" "6#%\"rG" }{TEXT -1 11 " apply the " }{XPPEDIT 18 0 "evalf;" "6#%&evalfG" }{TEXT -1 9 " command." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 9 "evalf(r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+K8&3R(!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 32 "Compare the following two cases." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "solve(exp(x )=y, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#%\"yG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "Yes, it \+ is well known that the inverse of the exponential function is the natu ral logarithm function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "solve(x*exp(x)=y, x);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%)LambertWG6#%\"yG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 131 "This function is not so \+ well known (yet). Until recent years, the latter result would have be en expressed in the implicit notation" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "implicit_soln := RootOf (_Z*exp(_Z)-y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.implicit_solnG-% 'RootOfG6#,&*&%#_ZG\"\"\"-%$expG6#F*F+F+%\"yG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "and we could mani pulate this form; for example," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "series(implicit_soln, y);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#+/%\"yG\"\"\"\"\"\"!\"\"\"\"##\"\"$\" \"#\"\"$#!\")F*\"\"%#\"$D\"\"#C\"\"&-%\"OG6#F%\"\"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "which agrees with the series solution of the ex plicit function known as " }{TEXT 285 8 "LambertW" }{TEXT -1 1 ":" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "series(LambertW(y), y);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#+/%\"yG\"\"\"\"\"\"!\"\"\"\"##\"\"$\" \"#\"\"$#!\")F*\"\"%#\"$D\"\"#C\"\"&-%\"OG6#F%\"\"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{XPPEDIT 18 0 "LambertW;" "6#%)LambertWG" }{TEXT -1 85 " function is defined by the above equation; i.e. it is defined by the property that " } {XPPEDIT 18 0 "W(y)*exp(W(y)) = y" "6#/*&-%\"WG6#%\"yG\"\"\"-%$expG6#- F&6#F(F)F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 81 "For detaile d information see [Corless96], or in a Maple session see the help page " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "?LambertW" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "There are many equations whose roots can be expressed in terms of the " }{TEXT 286 8 "LambertW" }{TEXT -1 37 " function. Here is one more example." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "eqn := ln(cos(2*Pi*x)^2/(y^2+1)) = y*cos(2*Pi*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/-%#lnG6#*&*$)-%$cosG6#,$*&%#PiG\"\"\"% \"xGF2\"\"#F4\"\"\"F5,&*$)%\"yGF4F5F2F2F2!\"\"*&F9F2F,F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "soln := solve(eqn, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solnG6$,$*&,&%#PiG\"\"\"-%'arccosG6#,$*&-%)L ambertWG6#,$*$-%%sqrtG6#*&)%\"yG\"\"#\"\"\",&*$F9F " 0 "" {MPLTEXT 1 0 67 "plot(soln[1], y=0..1); # This solution is yielding complex values." }}{PARA 8 "" 1 "" {TEXT -1 26 "Plotting error, empty plot" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "evalf( eval (soln[1], y=1/2) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%\"IG$\"+:Q/3 =!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot(soln[2], y=0.. 1);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVES G6$7V7$$\"+;arz@!#6$\"+tfDpZ!#57$$\"+\"y%*z7$F*$\"+w.wDZF-7$$\"+XTFwSF *$\"+w\\S*o%F-7$$\"+pMrU^F*$\"+WPAaYF-7$$\"+\"z_\"4iF*$\"+(oEMi%F-7$$ \"+m6m#G(F*$\"+i/\"ef%F-7$$\"+S&phN)F*$\"+=/\"4d%F-7$$\"+:ddC%*F*$\"+ \\mN[XF-7$$\"+*=)H\\5F-$\"+#z%oFXF-7$$\"+[!3uC\"F-$\"+\"\\)\\$\\%F-7$$ \"+J$RDX\"F-$\"+V'fFY%F-7$$\"+)R'ok;F-$\"+u5-NWF-7$$\"+1J:w=F-$\"+3Ky5 WF-7$$\"+3En$4#F-$\"+qO'))Q%F-7$$\"+/RE&G#F-$\"+[purVF-7$$\"+D.&4]#F-$ \"+3qiaVF-7$$\"+vB_WhJ%F-7$$\"+347TLF-$\"+l$=_I%F-7$$\"+LY.KNF-$\"+pw3(H%F-7$$ \"+\"o7Tv$F-$\"+rg!))G%F-7$$\"+$Q*o]RF-$\"+I&QCG%F-7$$\"+\"=lj;%F-$\"+ 4**RwUF-7$$\"+V&R3beUF-7$ $\"+=>Y2aF-$\"+,U'pD%F-7$$\"+yXu9cF-$\"+]+sbUF-7$$\"+\\y))GeF-$\"+,?! \\D%F-7$$\"+i_QQgF-$\"+>#>XD%F-7$$\"+!y%3TiF-$\"+q`]aUF-7$$\"+O![hY'F- $\"+:4'[D%F-7$$\"+#Qx$omF-$\"+:.[bUF-7$$\"+u.I%)oF-$\"+v@UcUF-7$$\"+(p e*zqF-$\"+$p*\\dUF-7$$\"+C\\'QH(F-$\"+AV*)eUF-7$$\"+8S8&\\(F-$\"+)\\*Q gUF-7$$\"+0#=bq(F-$\"+F%=@E%F-7$$\"+2s?6zF-$\"+N8&RE%F-7$$\"+IXaE\")F- $\"+.%**fE%F-7$$\"+l*RRL)F-$\"+)=x!oUF-7$$\"+`<.Y&)F-$\"+f**GqUF-7$$\" +8tOc()F-$\"+]ZbsUF-7$$\"+\\Qk\\*)F-$\"+wNouUF-7$$\"+p0;r\"*F-$\"+zR;x UF-7$$\"+lxGp$*F-$\"+ZhSzUF-7$$\"+!oK0e*F-$\"+(R3=G%F-7$$\"+<5s#y*F-$ \"+lv5%G%F-7$$\"\"\"\"\"!$\"+rjc'G%F--%'COLOURG6&%$RGBG$\"#5!\"\"Fe[lF e[l-%+AXESLABELSG6$Q\"y6\"%!G-%%VIEWG6$;Fe[lFc[l%(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "evalf( eval (soln[2], y=1/2) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+=z(3E%!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "Find the point of minimum value of the solution on the interval " } {XPPEDIT 18 0 "0 .. 1;" "6#;\"\"!\"\"\"" }{TEXT -1 2 " ." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "deri v := diff(soln[2],y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&derivG,$*& ,&*&*&-%)LambertWG6#,$*$-%%sqrtG6#*&)%\"yG\"\"#\"\"\",&*$F3F6\"\"\"F9F 9F9F6#F9F5F9,&*&F4F9F7F6F5*$)F4\"\"$F6F5F9F6*(,&F9F9F*F9\"\"\")F4\"\"$ F6F7\"\"\"!\"\"F9*&F*F6*$)F4\"\"#F6FF!\"#F6*&-F06#,&F9F9*&*$)F*F5F6F6* $)F4\"\"#F6FF!\"%F6%#PiG\"\"\"FFF:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "fsolve(deriv=0, y, 0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+![Ut9'!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 54 "Example 3: Functions \+ Defined via the RootOf Construct" }}{PARA 0 "" 0 "" {TEXT -1 44 "Suppo se that we wish to solve the equation " }{XPPEDIT 18 0 "cos(2*x) = ta n(alpha*x/Pi);" "6#/-%$cosG6#*&\"\"#\"\"\"%\"xGF)-%$tanG6#*(%&alphaGF) F*F)%#PiG!\"\"" }{TEXT -1 8 " where " }{XPPEDIT 18 0 "alpha;" "6#%&al phaG" }{TEXT -1 29 " is a parameter in the range " }{XPPEDIT 18 0 "0 . . 1/2;" "6#;\"\"!*&\"\"\"\"\"\"\"\"#!\"\"" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "resta rt;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "eqn := cos(2*x) = ta n(alpha*x/Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/-%$cosG6#,$% \"xG\"\"#-%$tanG6#*&*&%&alphaG\"\"\"F*F2\"\"\"%#PiG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "soln := solve(eqn, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solnG,$-%'RootOfG6#,&*&%#_ZG\"\"\"%&alphaGF, F,*&-%'arctanG6#-%$cosG6#F+F,%#PiGF,!\"##F,\"\"#" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "Look at the graphs of " }{XPPEDIT 18 0 "cos(2*x);" "6#-%$cosG6#*&\"\"#\"\"\"%\"xGF(" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "tan(alpha*x/Pi);" "6#-%$tanG6#*(%&a lphaG\"\"\"%\"xGF(%#PiG!\"\"" }{TEXT -1 21 ", for some values of " } {XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 32 ", to see where the g raphs cross." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "leqn := lhs(eqn); reqn := rhs(eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%leqnG-%$cosG6#,$%\"xG\"\"#" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%%reqnG-%$tanG6#*&*&%&alphaG\"\"\"%\"xGF+\"\"\" %#PiG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "graph[0] := p lot( leqn, x=0..1, color=green ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "for k from 1 to 4 do\n graph[k] := plot( eval(reqn,a lpha=k/8), x=0..1, color=red )\nod:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plots[display](\{seq(graph[k],k=0..4)\});" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6)-%'CURVESG6$7S7$\"\"!$ \"\"\"F(7$$\"1nmm;arz@!#<$\"1R.@'=*\\!***!#;7$$\"1LL$e9ui2%F.$\"1I*>-Q 'ym**F17$$\"1nmm\"z_\"4iF.$\"1>$*=%)=*H#**F17$$\"1mmmT&phN)F.$\"16OR*f t1')*F17$$\"1LLe*=)H\\5F1$\"14ab\\;g!y*F17$$\"1nm\"z/3uC\"F1$\"1c9k,f&)F17$$\"1+++v'Hi#HF1$\"1B=?bDwN$)F17$$\"1nm\"z*e v:JF1$\"1-\"=osM/7)F17$$\"1LLL347TLF1$\"1cB*p(=B\\yF17$$\"1LLLLY.KNF1$ \"1h5/A1*pg(F17$$\"1++D\"o7Tv$F1$\"1+nH)[z7J(F17$$\"1LLL$Q*o]RF1$\"1)[ 6#yOZPqF17$$\"1++D\"=lj;%F1$\"122mL(oXs'F17$$\"1++vV&RJT!3'F17$$\"1LLeR\"3Gy%F1$\"1w1`dFLjd F17$$\"1nm;/T1&*\\F1$\"1`O!4DF8T&F17$$\"1mm\"zRQb@&F1$\"11M%3Tx`.&F17$ $\"1***\\(=>Y2aF1$\"1:z*G?;,q%F17$$\"1mm;zXu9cF1$\"1$R&4`BEIVF17$$\"1+ ++]y))GeF1$\"13i;(**f.%RF17$$\"1****\\i_QQgF1$\"1W`bO$=>b$F17$$\"1*** \\7y%3TiF1$\"1bn\\8%R,<$F17$$\"1****\\P![hY'F1$\"14v'3:j,u#F17$$\"1LLL $Qx$omF1$\"1ZTnu,0\\BF17$$\"1+++v.I%)oF1$\"1!)QKy$Gs#>F17$$\"1mm\"zpe* zqF1$\"1Zaci-(=a\"F17$$\"1+++D\\'QH(F1$\"1u'*ow=*y6\"F17$$\"1KLe9S8&\\ (F1$\"1*e-Gs#zqrF.7$$\"1***\\i?=bq(F1$\"1gSphA$)oHF.7$$\"1LLL3s?6zF1$! 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Q[EOt(\\F.7$F\\r$\"1P$>/m@JA&F.7$Far$\"10Vj*3S'zaF.7$Ffr$\"1j?n[tE:dF. 7$F[s$\"1Ugi)=%\\pfF.7$F`s$\"1S6Q3fkLiF.7$Fes$\"1tdN*>hOY'F.7$Fjs$\"1% 34_hg@r'F.7$F_t$\"1q\"F1Fbdl-F$6$7SF_[l7$F,$\"1)f+yuQ\"pMFg[l7$F3$\" 1uK-F.7$FL$\"1m&3(o+?7BF.7$FQ$\"1WS\\%y]+l#F.7$FV$\" 1!3X`%4odA`F.7$Fhp$\"1%\\$*HYOti&F.7$F]q$\"1^#f(Gf(>)fF.7$Fbq$\"1qBG5 o,'H'F.7$Fgq$\"1+8JB?rSmF.7$F\\r$\"1l\"ysN*3ppF.7$Far$\"10W$oGw=J(F.7$ Ffr$\"1Y%=?77oi(F.7$F[s$\"1(G:n&=omzF.7$F`s$\"1%HP:&[!*>$)F.7$Fes$\"1n jO`TbF')F.7$Fjs$\"1+7D)=1+'*)F.7$F_t$\"1U?VB#oOI*F.7$Fdt$\"1FNZ*Q&3S'* F.7$Fit$\"1L-&p8#zl**F.7$F^u$\"1Cv\\pzwK5F17$Fcu$\"1KB?&*zIl5F17$Fhu$ \"1@3Dwg2+6F17$F]v$\"1WDl8QgJ6F17$Fbv$\"1:X\\9u4m6F17$Fgv$\"1\"fox?y&) >\"F17$F\\w$\"1;V[ymbK7F17$Faw$\"1(RyT'R!eE\"F17$Ffw$\"1!ef*Q.k+8F17$F [x$\"1H)z9F@UL\"F17$F`x$\"1\"GE^B$fo8F17$Fex$\"1%=;&e>r-9F17$Fjx$\"1!G 3bw\"4M9F17$F_y$\"1%\\Xwm!4q9F17$Fdy$\"1vBR;,K-:F17$Fiy$\"1WP)*)4 " 0 "" {MPLTEXT 1 0 26 "H := unapply(soln, alpha);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"HGR6#%&alphaG6\"6$%)operatorG%&arrowGF(,$-%'RootOfG 6#,&*&%#_ZG\"\"\"9$F3F3*&-%'arctanG6#-%$cosG6#F2F3%#PiGF3!\"##F3\"\"#F (F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Numerically evaluate " } {XPPEDIT 18 0 "H(alpha);" "6#-%\"HG6#%&alphaG" }{TEXT -1 23 " for vari ous values of " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "for k from 1 to 5 do\n 'H'(k/10) = evalf(H(k/10))\nod;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"HG6##\"\"\"\"#5$\"+:K!4t(!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"HG6##\"\"\"\"\"&$\"+]xT6w!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"HG6##\"\"$\"#5$\"+vB>&\\(!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"HG6##\"\"#\"\"&$\"++H%>Q(!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"HG6##\"\"\"\"\"#$\"+lcUrs!#5" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 45 "Symbolic manipulation of the RootOf construct" }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "H(alpha);" "6#-%\" HG6#%&alphaG" }{TEXT -1 17 " defined via the " }{XPPEDIT 18 0 "RootOf " "6#%'RootOfG" }{TEXT -1 80 " construct can be not only evaluated, it can also be manipulated mathematically." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "For example, compute a series expa nsion." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "ser_H := series(H(alpha), alpha);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%&ser_HG+1%&alphaG,$%#PiG#\"\"\"\"\"%\"\"!#!\"\"\"\" )\"\"\",$*&\"\"\"F3F(!\"\"#F*\"#;\"\"#,$*&,&F/F**$)F(\"\"#F3F*F3*$)F( \"\"#F3F4#F.\"$c#\"\"$,$*&,&F=F*F;F*F3*$)F(\"\"$F3F4#F*\"$G\"\"\"%,$*& ,(FKF*F;\"$g\"*$)F(F+F3\"\"$F3*$)F(\"\"%F3F4#F.\"&%Q;\"\"&-%\"OG6#F*\" \"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Perhaps we wish to know the rate of change of the solution: " }{XPPEDIT 18 0 "diff(H(alpha),alpha);" "6#-%%diffG6$-%\"HG6#%&alphaGF) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "diff_H := diff(H(alpha), alpha);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'diff_HG,$*&-%'RootOfG6#,&*&%#_ZG\"\"\"%&alphaGF-F-*& -%'arctanG6#-%$cosG6#F,F-%#PiGF-!\"#\"\"\",&F.F-*&*&-%$sinG6#F'F-F6F8F 8,&F-F-*$)-F4F>\"\"#F8F-!\"\"FCFD#!\"\"FC" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "or, for small values of \+ " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 67 " the derivative i s well approximated by differentiating the series:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "diff(ser_H, alpha);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+/%&alphaG#!\"\"\"\")\"\"! ,$*&\"\"\"F+%#PiG!\"\"#\"\"\"F'\"\"\",$*&,&F'F/*$)F,\"\"#F+F/F+*$)F,\" \"#F+F-#!\"$\"$c#\"\"#,$*&,&F6F/F4F/F+*$)F,\"\"$F+F-#F/\"#K\"\"$,$*&,( \"$G\"F/F4\"$g\"*$)F,\"\"%F+\"\"$F+*$)F,\"\"%F+F-#!\"&\"&%Q;\"\"%-%\"O G6#F/\"\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "Note: We get the same series expansion by directly expand ing the expression " }{TEXT 290 6 "diff_H" }{TEXT -1 2 " :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "normal( ser ies(diff_H, alpha, 5) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+/%&alphaG #!\"\"\"\")\"\"!,$*&\"\"\"F+%#PiG!\"\"#\"\"\"F'\"\"\",$*&,&F'F/*$)F,\" \"#F+F/F+*$)F,\"\"#F+F-#!\"$\"$c#\"\"#,$*&,&F6F/F4F/F+*$)F,\"\"$F+F-#F /\"#K\"\"$,$*&,(\"$G\"F/F4\"$g\"*$)F,\"\"%F+\"\"$F+*$)F,\"\"%F+F-#!\"& \"&%Q;\"\"%-%\"OG6#F/\"\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 19 "Look at a graph of " }{XPPEDIT 18 0 "H(al pha);" "6#-%\"HG6#%&alphaG" }{TEXT -1 81 " by plotting the series appr oximation (first convert the series to a polynomial)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "poly_H : = convert(ser_H, polynom);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'poly_ HG,.%#PiG#\"\"\"\"\"%%&alphaG#!\"\"\"\")*&*$)F*\"\"#\"\"\"F2F&!\"\"#F( \"#;*&*&,&F-F(*$)F&F1F2F(F()F*\"\"$F2F2*$)F&\"\"#F2F3#F,\"$c#*&*&,&F1F (F9F(F()F*F)F2F2*$)F&\"\"$F2F3#F(\"$G\"*&*&,(FJF(F9\"$g\"*$)F&F)F2F " 0 "" {MPLTEXT 1 0 27 "plot(poly_H, alpha=0..1/2);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7S7$\"\"!$\"1$[uRj\") R&y!#;7$$\"1LLL3x&)*3\"!#<$\"1MD\"=&>QSyF+7$$\"1nm\"H2P\"Q?F/$\"1xp9op eGyF+7$$\"1LL$eRwX5$F/$\"1^?*)*3k`\"yF+7$$\"1LL$3x%3yTF/$\"1mhY*y(4-yF +7$$\"1mm\"z%4\\Y_F/$\"1/#=36Q*)y(F+7$$\"1LLeR-/PiF/$\"1nMC;exwxF+7$$ \"1***\\il'pisF/$\"1\"y1B+@Uw(F+7$$\"1LLe*)>VB$)F/$\"1'3X%3wF^xF+7$$\" 1***\\7`l2Q*F/$\"1LP.'e;%QxF+7$$\"1nm;/j$o/\"F+$\"1M)*4^&H_s(F+7$$\"1L L3_>jU6F+$\"1u#ezS\\Or(F+7$$\"1++]i^Z]7F+$\"1/X#Rt^1q(F+7$$\"1++](=h(e 8F+$\"1.IQS;k(o(F+7$$\"1++]P[6j9F+$\"1!*yEBC9vwF+7$$\"1L$e*[z(yb\"F+$ \"1'[t)yU#Qm(F+7$$\"1nm;a/cq;F+$\"1b'=Ed0/l(F+7$$\"1nmm;t,mb#fi(F+7$$\"1mmm\"pW`(>F+$\"1([Jg$HK9wF +7$$\"1+]i!f#=$3#F+$\"1vXx*HH;g(F+7$$\"1+](=xpe=#F+$\"1UY\"Q&od*e(F+7$ $\"1nm\"H28IH#F+$\"13*)3>t.xvF+7$$\"1n;zpSS\"R#F+$\"1W,a;TblvF+7$$\"1L L3_?`(\\#F+$\"1E;wY>?`vF+7$$\"1L$e*)>pxg#F+$\"1&eGa_3/a(F+7$$\"1+]Pf4t .FF+$\"1:;X?AIHvF+7$$\"1LLe*Gst!GF+$\"1Ih;*eQt^(F+7$$\"1+++DRW9HF+$\"1 Q70&*H,0vF+7$$\"1***\\7j#>>IF+$\"1&4zK#z)H\\(F+7$$\"1+]i!RU07$F+$\"1V \\eARQ\"[(F+7$$\"1++v=S2LKF+$\"1Z:[#)[`ouF+7$$\"1mmm\"p)=MLF+$\"1y&*\\ b2-duF+7$$\"1++](=]@W$F+$\"1#>KGNfZW(F+7$$\"1L$e*[$z*RNF+$\"1Sa54tnLuF +7$$\"1++]iC$pk$F+$\"1EfjHDf@uF+7$$\"1m;H2qcZPF+$\"1mfk63D5uF+7$$\"1+] 7.\"fF&QF+$\"1NY8J`U)R(F+7$$\"1mm;/OgbRF+$\"17rj0I*oQ(F+7$$\"1+]ilAFjS F+$\"1%zV/S][P(F+7$$\"1LLL$)*pp;%F+$\"1qlE/8GjtF+7$$\"1LL3xe,tUF+$\"1! \\&>H(z9N(F+7$$\"1n;HdO=yVF+$\"1$egp.0)RtF+7$$\"1+++D>#[Z%F+$\"1#3pRb- \"HtF+7$$\"1nmT&G!e&e%F+$\"18;w^e'oJ(F+7$$\"1LLL$)Qk%o%F+$\"1=sDqw%fI( F+7$$\"1+]iSjE!z%F+$\"1W3PcQL%H(F+7$$\"1+]P40O\"*[F+$\"1U\"y0-WKG(F+7$ $\"1+++++++]F+$\"1x2t%fa8F(F+-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLA BELSG6$Q&alpha6\"%!G-%%VIEWG6$;F($\"+++++]!#5%(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 40 "Final remar ks about the RootOf construct" }}{PARA 0 "" 0 "" {TEXT -1 25 "We defin ed the function " }{XPPEDIT 18 0 "H(alpha) = RootOf(_Z*alpha-2*arctan (cos(_Z))*Pi)/2;" "6#/-%\"HG6#%&alphaG*&-%'RootOfG6#,&*&%#_ZG\"\"\"F'F /F/*(\"\"#F/-%'arctanG6#-%$cosG6#F.F/%#PiGF/!\"\"F/\"\"#F9" }{TEXT -1 3 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Is the function " }{XPPEDIT 18 0 "H(alpha);" "6#-%\"HG6#%&alphaG" } {TEXT -1 24 " a closed form solution?" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 29 "In the traditional sense, no." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "But in a \+ computer algebra system, when we have given a name (such as " } {XPPEDIT 18 0 "H;" "6#%\"HG" }{TEXT -1 158 ") to a function and, more \+ importantly, when we can perform mathematical manipulations (symbolic \+ as well as numerical) with this function, then is it really a " } {TEXT 287 11 "lower class" }{TEXT -1 10 " function?" }}{PARA 0 "" 0 " " {TEXT -1 145 "Is it less useful than a solution expressed in terms o f functions which have previously been given names in the mathematical literature, such as " }{TEXT 291 6 "arctan" }{TEXT -1 5 ", or " } {TEXT 289 9 "EllipticK" }{TEXT -1 46 ", or a more recent entry into th e literature, " }{TEXT 288 8 "LambertW" }{TEXT -1 2 " ?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 28 "Closed-Form Solution of ODEs" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 23 "Functionality of dsol ve" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{MPLTEXT 0 21 6 "dsolve" } {TEXT -1 86 " command is a general ODE solver which handles various ty pes of ODE problems including" }}{PARA 15 "" 0 "" {TEXT -1 56 "closed- form solution of a single ODE or a system of ODEs" }}{PARA 15 "" 0 "" {TEXT -1 70 "closed-form solution of ODEs with given initial or bounda ry conditions" }}{PARA 15 "" 0 "" {TEXT -1 73 "formal power series sol ution of a linear ODE with polynomial coefficients" }}{PARA 15 "" 0 " " {TEXT -1 42 "solutions obtained via integral transforms" }}{PARA 15 "" 0 "" {TEXT -1 26 "truncated series solutions" }}{PARA 15 "" 0 "" {TEXT -1 19 "numerical solutions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 0 {PARA 5 "" 0 "" {TEXT -1 36 "Solution Techniques for a Single \+ ODE" }}{PARA 0 "" 0 "" {TEXT -1 48 "For a single ODE there are two bas ic strategies." }}{PARA 15 "" 0 "" {TEXT 269 22 "Classification method s" }}{PARA 16 "" 0 "" {TEXT -1 62 "determine whether the given ODE mat ches a recognizable pattern" }}{PARA 16 "" 0 "" {TEXT -1 54 "if yes, a pply a known solution method for that pattern" }}{PARA 16 "" 0 "" {TEXT -1 71 "Special Case: Linear ODE, for which decision procedures a re implemented" }}{PARA 15 "" 0 "" {TEXT 270 16 "Symmetry methods" }} {PARA 16 "" 0 "" {TEXT -1 59 "look for generators of the symmetry grou ps of the given ODE" }}{PARA 16 "" 0 "" {TEXT -1 85 "use this informat ion to integrate the ODE, or at least to reduce the order of the ODE" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 22 "The odeadvisor command" }} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 267 10 "odeadvisor" }{TEXT -1 43 " command is an innovative addition to Maple" }}{PARA 15 "" 0 " " {TEXT -1 36 "especially for teaching and learning" }}{PARA 15 "" 0 " " {TEXT -1 69 "classifies a given ODE according to standard textbook c lassifications" }}{PARA 15 "" 0 "" {TEXT -1 66 "provides user access t o the classification strategies used by the " }{TEXT 268 6 "dsolve" } {TEXT -1 8 " command" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 20 "Examples : Single ODE" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools ):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "ode[1] : = x*diff(y(x),x)+a*y(x)^2-y(x)+b*x^2;" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$odeG6#\"\"\",**&%\"xGF'-%%diffG6$-%\"yG6#F*F*F'F '*&%\"aGF')F.\"\"#\"\"\"F'F.!\"\"*&%\"bGF')F*F4F5F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Ask for informa tion about the ODE." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "odeadvisor(o de[1]);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%7$%-_homog eneousG%(class~DG%*_rationalG%)_RiccatiG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Solve the ODE." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dsolve(ode[1], y(x));" }{TEXT -1 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&*(-%$tanG6#,&*&F'\"\" \"-%%sqrtG6#*&%\"bGF/%\"aGF/\"\"\"!\"\"*&%$_C1GF/F0F6F7F/F'F6F0F6F6F5! \"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "ode[2] : = (2*y(x)-x)*diff(y(x),x)-y(x)-2*x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>&%$odeG6#\"\"#,(*&,&-%\"yG6#%\"xGF'F.!\"\"\"\"\"-%%diffG6$F+F.F0F0F+ F/F.!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "odeadvisor(ode[ 2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&7$%-_homogeneousG%(class~AG% '_exactG%*_rationalG7%%&_AbelG%)2nd~typeGF&" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dsolve(ode[2], y( x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"yG6#%\"xG,&F'#\"\"\"\"\"# *$-%%sqrtG6#,&*$)F'F+\"\"\"\"\"&*$)-%$expG6#%$_C1GF+F3\"\"%F3F)/F$,&F' F)F,#!\"\"F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 " " {TEXT -1 30 "Example 3: A second-order ODE" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "ode[3] := x^2*diff(y(x),x,x)+x*diff(y(x),x)-(x^2 +n^2)*y(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$odeG6#\"\"$,(*&)%\" xG\"\"#\"\"\"-%%diffG6$-%\"yG6#F+-%\"$G6$F+F,\"\"\"F7*&F+F7-F/6$F1F+F7 F7*&,&*$F*F-F7*$)%\"nGF,F-F7F7F1F7!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "odeadvisor(ode[3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#7#7$%(_BesselG%*_modifiedG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dsolve(ode[3], y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&%$_C1G\"\"\"-%(BesselKG6$% \"nGF'F+F+*&%$_C2GF+-%(BesselIGF.F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "info" {TEXT -1 24 "The Clas sification Types" }}{PARA 0 "" 0 "" {TEXT -1 16 "The goal of the " } {TEXT 259 10 "odeadvisor" }{TEXT -1 125 " command is to classify a giv en ODE according to standard textbooks (cf. [Zwillinger92]). The clas sification types known to " }{TEXT 258 10 "odeadvisor" }{TEXT -1 16 " \+ are as follows." }}{PARA 15 "" 0 "" {TEXT -1 16 "First order ODEs" }} {PARA 17 "" 0 "" {HYPERLNK 17 "Abel" 2 "odeadvisor[Abel]" "" }{TEXT -1 11 ", " }{HYPERLNK 17 "Abel2A" 2 "odeadvisor[Abel2A]" "" } {TEXT -1 9 ", " }{HYPERLNK 17 "Abel2C" 2 "odeadvisor[Abel2C]" " " }{TEXT -1 9 ", " }{HYPERLNK 17 "Bernoulli" 2 "odeadvisor[Bern oulli]" "" }{TEXT -1 5 ", " }{HYPERLNK 17 "Chini" 2 "odeadvisor[Chi ni]" "" }{TEXT -1 10 ", \n" }{HYPERLNK 17 "Clairaut" 2 "odeadvi sor[Clairaut]" "" }{TEXT -1 7 ", " }{HYPERLNK 17 "dAlembert" 2 "o deadvisor[dAlembert]" "" }{TEXT -1 6 ", " }{HYPERLNK 17 "exact" 2 "odeadvisor[exact]" "" }{TEXT -1 10 ", " }{HYPERLNK 17 "homoge neous" 2 "odeadvisor[homogeneous]" "" }{TEXT -1 3 ", " }{HYPERLNK 17 "homogeneousB" 2 "odeadvisor[homogeneousB]" "" }{TEXT -1 3 ", \n" } {HYPERLNK 17 "homogeneousC" 2 "odeadvisor[homogeneousC]" "" }{TEXT -1 3 ", " }{HYPERLNK 17 "homogeneousD" 2 "odeadvisor[homogeneousD]" "" } {TEXT -1 3 ", " }{HYPERLNK 17 "homogeneousG" 2 "odeadvisor[homogeneou sG]" "" }{TEXT -1 3 ", " }{HYPERLNK 17 "linear" 2 "odeadvisor[linear] " "" }{TEXT -1 8 ", " }{HYPERLNK 17 "patterns" 2 "odeadvisor[pat terns]" "" }{TEXT -1 7 ", \n" }{HYPERLNK 17 "quadrature" 2 "odeadv isor[quadrature]" "" }{TEXT -1 5 ", " }{HYPERLNK 17 "rational" 2 "o deadvisor[rational]" "" }{TEXT -1 7 ", " }{HYPERLNK 17 "Riccati" 2 "odeadvisor[Riccati]" "" }{TEXT -1 8 ", " }{HYPERLNK 17 "separ able" 2 "odeadvisor[separable]" "" }{TEXT -1 5 ", " }{HYPERLNK 17 " sym_implicit" 2 "odeadvisor[sym_implicit]" "" }{TEXT -1 3 " " }} {PARA 15 "" 0 "" {TEXT -1 17 "Second order ODEs" }}{PARA 17 "" 0 "" {HYPERLNK 17 "Bessel" 2 "odeadvisor[Bessel]" "" }{TEXT -1 7 ", " }{HYPERLNK 17 "Duffing" 2 "odeadvisor[Duffing]" "" }{TEXT -1 8 ", \+ " }{HYPERLNK 17 "ellipsoidal" 2 "odeadvisor[ellipsoidal]" "" }{TEXT -1 7 ", " }{HYPERLNK 17 "elliptic" 2 "odeadvisor[elliptic]" "" } {TEXT -1 5 ", " }{HYPERLNK 17 "Emden" 2 "odeadvisor[Emden]" "" } {TEXT -1 8 ", \n" }{HYPERLNK 17 "erf" 2 "odeadvisor[erf]" "" } {TEXT -1 10 ", " }{HYPERLNK 17 "exact_linear" 2 "odeadvisor[ex act_linear]" "" }{TEXT -1 3 ", " }{HYPERLNK 17 "exact_nonlinear" 2 "o deadvisor[exact_nonlinear]" "" }{TEXT -1 3 ", " }{HYPERLNK 17 "Gegenb auer" 2 "odeadvisor[Gegenbauer]" "" }{TEXT -1 3 ", " }{HYPERLNK 17 "H alm" 2 "odeadvisor[Halm]" "" }{TEXT -1 9 ", \n" }{HYPERLNK 17 "H ermite" 2 "odeadvisor[Hermite]" "" }{TEXT -1 6 ", " }{HYPERLNK 17 "Jacobi" 2 "odeadvisor[Jacobi]" "" }{TEXT -1 9 ", " }{HYPERLNK 17 "Lagerstrom" 2 "odeadvisor[Lagerstrom]" "" }{TEXT -1 8 ", " } {HYPERLNK 17 "Laguerre" 2 "odeadvisor[Laguerre]" "" }{TEXT -1 5 ", \+ " }{HYPERLNK 17 "Lienard" 2 "odeadvisor[Lienard]" "" }{TEXT -1 6 ", \+ \n" }{HYPERLNK 17 "Liouville" 2 "odeadvisor[Liouville]" "" }{TEXT -1 4 ", " }{HYPERLNK 17 "linear_ODEs" 2 "odeadvisor[linear_ODEs]" "" } {TEXT -1 4 ", " }{HYPERLNK 17 "linear_sym" 2 "odeadvisor[linear_sym] " "" }{TEXT -1 8 ", " }{HYPERLNK 17 "missing" 2 "odeadvisor[miss ing]" "" }{TEXT -1 6 ", " }{HYPERLNK 17 "Painleve" 2 "odeadvisor[P ainleve]" "" }{TEXT -1 5 ", \n" }{HYPERLNK 17 "quadrature" 2 "odeadv isor[quadrature]" "" }{TEXT -1 3 ", " }{HYPERLNK 17 "reducible" 2 "od eadvisor[reducible]" "" }{TEXT -1 6 ", " }{HYPERLNK 17 "sym_Fx" 2 "odeadvisor[sym_Fx]" "" }{TEXT -1 12 ", " }{HYPERLNK 17 "Tit chmarsh" 2 "odeadvisor[Titchmarsh]" "" }{TEXT -1 3 ", " }{HYPERLNK 17 "Van_der_Pol" 2 "odeadvisor[Van_der_Pol]" "" }{TEXT -1 2 " " }} {PARA 15 "" 0 "" {TEXT -1 15 "High order ODEs" }}{PARA 17 "" 0 "" {HYPERLNK 17 "quadrature" 2 "odeadvisor[quadrature]" "" }{TEXT -1 3 ", " }{HYPERLNK 17 "missing" 2 "odeadvisor[missing]" "" }{TEXT -1 3 ", \+ " }{HYPERLNK 17 "exact_linear" 2 "odeadvisor[exact_linear]" "" } {TEXT -1 3 ", " }{HYPERLNK 17 "exact_nonlinear" 2 "odeadvisor[exact_n onlinear]" "" }{TEXT -1 3 ", " }{HYPERLNK 17 "reducible" 2 "odeadviso r[reducible]" "" }{TEXT -1 2 ", " }}{PARA 17 "" 0 "" {HYPERLNK 17 "lin ear_ODEs" 2 "odeadvisor[linear_ODEs]" "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "For further information see the help page " }{TEXT 260 11 "?odeadvisor" }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 44 "Example 4: Sol ving via Lie symmetry methods" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "ode[4] := diff(y(x),x)=F((y(x)-x*ln(x))/x) + ln(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$odeG6#\"\"%/-%%diffG6$-%\"yG6#%\"xGF/,&-%\" FG6#*&,&F,\"\"\"*&F/F6-%#lnGF.F6!\"\"\"\"\"F/!\"\"F6F8F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "Lie symme try methods are used to express the solution, in this case in terms of Maple's " }{XPPEDIT 18 0 "RootOf;" "6#%'RootOfG" }{TEXT -1 11 " const ruct." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dsolve(ode[4], y(x ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&-%'RootOfG6#, (-%#lnGF&\"\"\"-%$IntG6$*&\"\"\"F5,(%#_aGF0F0F0-%\"FG6#F7!\"\"!\"\"/F7 ;%!G%#_ZGF0%$_C1GF;F0F'F0F0*&F'F5F.F0F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "For the following particu lar choice of the function " }{TEXT 257 1 "F" }{TEXT -1 47 ", the abov e ODE is an equation of Ricatti type." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "F := u -> u^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"FGR6#%\"uG6\"6$%)operatorG%&arrowGF(*$)9$\"\"#\"\"\"F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "ode[4];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"yG6#%\"xGF*,&*&*$),&F'\"\"\"*&F*F0-%#lnG F)F0!\"\"\"\"#\"\"\"F6*$)F*\"\"#F6!\"\"F0F2F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "odeadvisor(ode[4], y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#%)_RiccatiG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "Request that Lie symmetry methods be us ed to compute the solution." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "dsolve(ode[4], y(x), 'Lie');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %\"yG6#%\"xG,$*(F'\"\"\",(*&-%%sqrtG6#\"\"&\"\"\"-%#lnGF&F*\"\"#*$F-F1 F*-%%tanhG6#,$*&F-F1,&F2!\"\"%$_C1GF*F*#F*F4F0F*F-F1#F*\"#5" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "Sh ow the symmetry generator (a set of infinitesimals which leave the ODE invariant)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "symgen(ode[ 4]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$/%$_xiG%\"xG/%%_etaG,&F&\" \"\"%\"yGF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 19 "The DEtools package" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 271 7 "DEtools" }{TEXT -1 61 " package provid es user tools for manipulating ODEs, including" }}{PARA 15 "" 0 "" {TEXT -1 27 "functions for visualization" }}{PARA 16 "" 0 "" {TEXT -1 4 "e.g." }{TEXT 273 34 " DEplot, dfieldplot, phaseportrait" }}{PARA 15 "" 0 "" {TEXT -1 46 "general manipulation of differential equations " }}{PARA 16 "" 0 "" {TEXT -1 5 "e.g. " }{TEXT 272 47 "Dchangevar, con vertsys, indicialeq, reduceOrder" }}{PARA 15 "" 0 "" {TEXT -1 48 "func tions for constructing closed form solutions" }}{PARA 16 "" 0 "" {TEXT -1 5 "e.g. " }{TEXT 274 67 "constcoeffsols, exactsol, expsols, l inearsol, ratsols, separablesol" }}{PARA 15 "" 0 "" {TEXT -1 36 "funct ions for differential operators" }}{PARA 16 "" 0 "" {TEXT -1 5 "e.g. \+ " }{TEXT 275 28 "DFactor, adjoint, formal_sol" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 22 "Visualization Ex amples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Using " } {TEXT 276 6 "DEplot" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "de1 := cos(x)*diff(y(x),x$3 )-diff(y(x),x$2)+Pi*diff(y(x),x)=y(x)-x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$de1G/,(*&-%$cosG6#%\"xG\"\"\"-%%diffG6$-%\"yGF*-%\"$G6$F+\"\" 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Cheb-Terra b, " }{TEXT 294 38 "Maple worksheet on dsolve in education" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "referenced in the web page " } {TEXT 292 47 "http://daisy.uwaterloo.ca/~kogeddes/papers.html" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 131 "Note that it is possible to \+ view this Maple worksheet in a Maple V session (but to re-execute the \+ worksheet would require maple 6)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 19 "The DESol Construct" }}{PARA 0 " " 0 "" {TEXT -1 21 "Corresponding to the " }{XPPEDIT 18 0 "RootOf;" "6 #%'RootOfG" }{TEXT -1 82 " construct for algebraic expressions, discus sed in a previous section, there is a " }{XPPEDIT 18 0 "DESol;" "6#%&D ESolG" }{TEXT -1 143 " construct in Maple for the implicit representat ion of solutions of a differential equation. For details, in a Maple \+ session see the help page" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " ?DESol" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}}{SECT 0 {PARA 3 " " 0 "" {TEXT -1 26 "Numerical Solution of ODEs" }}{EXCHG {PARA 4 "" 0 "" {TEXT -1 18 "A Ricatti equation" }}{PARA 0 "" 0 "" {TEXT -1 61 "Con sider the Ricatti equation ode[4] from a previous section." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "ode[4] := diff(y(x),x) = (y(x) - x* ln(x))^2/x^2 + ln(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$odeG6#\" \"%/-%%diffG6$-%\"yG6#%\"xGF/,&*&*$),&F,\"\"\"*&F/F5-%#lnGF.F5!\"\"\" \"#\"\"\"F;*$)F/\"\"#F;!\"\"F5F7F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "with initial condition" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "ic := y(1)=1/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG 6#\"\"\"#F)\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "soln := dsolve(\{ode[4],ic\}, y(x), 'Lie'):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "exact_ans := rhs(soln);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*exact_ansG,$*(%\"xG\"\"\",(*&-%%sqrtG6#\"\"&\"\"\"-%#lnG6#F'F (\"\"#*$F+F/F(-%%tanhG6#,$F*#F(F3!\"&F(F+F/#F(\"#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Standard numerical solution." }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 55 "numeric_ans := dsolve(\{ode[4],ic\}, y(x), t ype=numeric);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,numeric_ansGR6#%(r kf45_xG6'%\"iG%(rkf45_sG%)outpointG%#r1G%#r2G6#%aoCopyright~(c)~1993~b y~the~University~of~Waterloo.~All~rights~reserved.G6\"C&>8&-%&evalfG6# 9$@$52-%$absG6#,&$\"\"\"\"\"!F@F3!\"\"-F<6#,&&%,loc_controlG6#\"\"#F@F 3FB4-%'memberG6$&FG6#\"\"'<*$!\"#FA$FBFAFBF@FSFI$FIFAF?C%>FG-%%copyG6# =F06#;F@\"#EE\\[l;\"#FA\"\")\"&++$F@F@\"#6FA\"#=FA \"#9FA\"#:FA\"#DFA\"#@FA\"#CFA\"\"&$F@!\")\"\"($F@!\"*\"\"%Fho\"#;FA\" #?$\"+++++]!#5\"#7FAFIF?\"\"$F?\"#BFAFPF@\"#5FAFhnFA\"\"*\"%+5>%'loc_y 0G-FY6#=F06#;F@F@E\\[l\"F@F`p>%'loc_y1G-FY6#=F0F^qE\\[l!@$0F;FAC$>&FG6 #FdpF3@%1%'DigitsG-%'evalhfG6#F_rC$>8%-%*traperrorG6#-Far6#-%=dsolve/n umeric_solnall_rkf45G6,%&loc_FG-%$varG6#FG-F`s6#Fjp-F`s6#Fbq-F`s6#%'lo c_F1G-F`s6#%'loc_F2G-F`s6#%'loc_F3G-F`s6#%'loc_F4G-F`s6#%'loc_F5G-F`s6 #%)loc_workG@$/Fer%*lasterrorGC%>8'-%+searchtextG6$.Far-%(convertG6$-% #opG6$F@7#Fer%%nameG>8(-F_u6$.%)hardwareGFbu@%50F]uFA0F[vFA-F\\s6,F^sF GFjpFbqFhsF[tF^tFatFdtFgt-%&ERRORG6#FerFdv7$/%\"xGF7-%$seqG6$/&%$ordG6 #,&8$F@F@F@&Fjp6#Fdw/FdwF_qF06%FGFjpFbqF0" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 17 "numeric_ans(1.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#7$/%\"xG$\"#:!\"\"/-%\"yG6#F%$\"1!4'eb.wgk!#;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 17 "numeric_ans(4.0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$/%\"xG$\"#S!\"\"/-%\"yG6#F%$\"1O+kk/meM!#:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Compare with the exact answer." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "evalf(subs(x=4,exact_ans), \+ 15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0_xSUg'eM!#9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 38 "Various numerical ODE solvers i n Maple" }}{PARA 0 "" 0 "" {TEXT -1 80 "The following numerical method s have been implemented in Maple for solving ODEs." }}{PARA 15 "" 0 " " {TEXT -1 77 "classical (i.e. various classical methods, implemented \+ for teaching purposes)" }}{PARA 15 "" 0 "" {TEXT -1 5 "rkf45" }}{PARA 15 "" 0 "" {TEXT -1 7 "dverk78" }}{PARA 15 "" 0 "" {TEXT -1 5 "lsode" }}{PARA 15 "" 0 "" {TEXT -1 4 "gear" }}{PARA 15 "" 0 "" {TEXT -1 5 "mg ear" }}{PARA 15 "" 0 "" {TEXT -1 12 "taylorseries" }}{PARA 0 "" 0 "" {TEXT -1 31 "For details, see the help page " }{TEXT 261 16 "?dsolve[n umeric]" }{TEXT -1 31 " and its associated help pages." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 21 "Example usin g dverk78" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "numeric_ans := dsolve(\{ode[4],ic\}, y(x), type=numeric, method=dverk78);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,numeric_ansGR6#%*dverk78_tG6$%\"iG%*dverk 78_sG6#%aoCopyright~(c)~1993~by~the~University~of~Waterloo.~All~rights ~reserved.G6\"C$@%1%'DigitsG-%'evalhfG6#F1C$>8%-%*traperrorG6#-F36#-%@ dsolve/numeric/dverk78_rdmeth2bG6,9$%$fcnG-%$varG6#%#y0G%:dsolve/numer ic/dverk78_aaG%:dsolve/numeric/dverk78_bbG%:dsolve/numeric/dverk78_ddG -FC6#%#c0G-FC6#%#y1G-FC6#%#y2G-FC6#%\"wG@$/F7%*lasterrorG@%2&FK6#\"#H \"\"!-%&ERRORG6#F7-F>6,F@FAFEFFFGFHFKFNFQFTF[o7$/%\"xGF@-%$seqG6$/&%(d verk78G6#,&8$\"\"\"FioFio&FQ6#Fho/Fho;FioFioF-F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "numeric_ans(1.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$/%\"xG$\"#:!\"\"/-%\"yG6#F%$\"1uiBw+wgk!#;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "numeric_ans(4.0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$/%\"xG$\"#S!\"\"/-%\"yG6#F%$\"1'4kSUg'eM!# :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Compare with the exact answe r." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "evalf(subs(x=4,exact_ ans), 15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0_xSUg'eM!#9" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 29 "The Interpolant as a Solu tion" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 138 "Rather than a procedure \+ which invokes the numerical method for the solution at any specified p oint, it would be more useful to return the " }{TEXT 256 21 "piecewise interpolant" }{TEXT -1 37 " representing the numerical solution." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "This concept is being investigated for a future version of Mapl e." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "numeric_ans := dsolve(\{ode[4],ic\}, y(x), type=numer ic,\n method=dverk78, value=array([4])):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 159 "In the research version of Maple, the piecewis e interpolant (which is constructed by the dverk78 numerical method in any case) can be returned as the solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "For the above example, we would get the following piecewise function as the solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 12 "" 1 "" {XPPMATH 20 "6#>%.piecew ise_ansG-%*PIECEWISEG6+7$\"\"!2%\"xG$\"\"\"F)7$,4*$)F+\"\")\"\"\"$!++] P%)R!#6*$)F+\"\"(F3$\"+&)*\\(=nF6*$)F+\"\"'F3$\"*)\\Pf%*!\"**$)F+\"\"& F3$!+.+vn\\FA*$)F+\"\"%F3$\"++]g!3\"!\")*$)F+\"\"$F3$!+xhK-7FL*$)F+\" \"#F3$\"+g+h,jFAF+$!(C2'\\!\"($!*m9dQ*!#5F-2F+$\"+++++7FA7$,4F0$\"+$p` TW$F6F7$!+B3mkUFfnF<$\"+QdwcAFAFB$!+<(*G*f'FAFG$\"+&z;r8\"FLFM$!+bwc16 FLFR$\")zqx\\FYF+$\"**)p`s\"FA$!*c#)=@#FAF-2F+$\"+.aE'\\\"FA7$,4F0$\"+ 7'*)yH\"!#7F7$!+H'Q-A\"F6F<$\"*e8n_\"FfnFB$\"+i4%*\\FFfnFG$!*(ocDFA7$,4F 0$!+[mo49FepF7$\"+w@FnEF6F<$!+(3c[E#FfnFB$\"+AT9F6FAFG$!+S*y$*e$FAFM$ \"*yrZU(FLFR$!)&QYG*FYF+$\"*!or`kFL$!*(yYI9FLF-2F+$\"+rFepF7$\"+4r!*pbF6F<$!+LxekoFfnFB$\"+mWSQ[FAFG$!+%ewE8#FLFM$\"+%**f^ ,'FLFR$!+3Efc5FYF+$\"+)QD91\"FY$!+7H9GYFLF-2F+$FIF)7$F)1FdwF+" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "It is easy to manipulate the piecewise solution, e.g. " }{TEXT 262 13 "d ifferentiate" }{TEXT -1 2 ", " }{TEXT 263 9 "integrate" }{TEXT -1 9 ", ... or " }{TEXT 264 4 "plot" }{TEXT -1 4 " it." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "plot([piece wise_ans, exact_ans], x=0.5..4.5,\n style=[line,point], color=[blue ,red]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CU RVESG6%7co7$$\"1+++++++]!#;\"\"!7$$\"1nmmmh)=(eF*F+7$$\"1LLLe'40j'F*F+ 7$$\"1nmm;6m$[(F*F+7$$\"1nmm;yYU$)F*F+7$$\"1KLLeF>(>*F*F+7$$\"1*****\\ Z7Mf*F*F+7$$\"1mmm\">K'*)**F*F+7$$\"1L$3-OF:+\"!#:$\"1Z4?&Q:Q+&F*7$$\" 1++D,:4/5FD$\"1u0v*33-,&F*7$$\"1n;HUcl15FD$\"1_%[6E&e;]F*7$$\"1LLL$y># 45FD$\"1;CBiu%H-&F*7$$\"1nmTl![V,\"FD$\"1s(H+>Hc.&F*7$$\"1++]ZjZ>5FD$ \"1I`a;xD[]F*7$$\"1nmm6HtH5FD$\"1%pI(*psL2&F*7$$\"1LL$eZ*)*R5FD$\"1NG$ e*pK)4&F*7$$\"1nm;/E]g5FD$\"1)Ra9&z)y9&F*7$$\"1++]Kd,\"3\"FD$\"1;>e::? (>&F*7$$\"1LL$e9XM7\"FD$\"11J8b+Z*H&F*7$$\"1nm;fX(e;\"FD$\"1$QdhP\\PS& F*7$$\"1++]U7Y]7FD$\"1a]'H&=_CcF*7$$\"1LLLV!puL\"FD$\"17,kWpRzeF*7$$\" 1nmmhb599FD$\"192^1,jMhF*7$$\"1+++I,Q+:FD$\"1#='Rl3IikF*7$$\"1+++]*3qe \"FD$\"1))4I$)paQoF*7$$\"1+++q=\\q;FD$\"1@xl!yH%[sF*7$$\"1nm;fBIYm(F*7$$\"1LLLj$[k$=FD$\"1H\\)=SX_?)F*7$$\"1LLL`Q\"G\">FD$\" 1KrW9\\84()F*7$$\"1++]s]k,?FD$\"1AN[X05X$*F*7$$\"1LLL`dF!3#FD$\"1zA11' p;&**F*7$$\"1++]sgam@FD$\"1gpZ?bJm5FD7$$\"1++]>w9FD7$$\"1++]nZ )Hm#FD$\"1P*y\"RGsh:FD7$$\"1mmmJy*eu#FD$\"1zh6xdYd;FD7$$\"1+++S^bJGFD$ \"1(*[&>O?*fFD7$$\"1+++:#fk3$FD$\"1UKPB<([3#FD7$$\"1LLL`4NnJFD$\"1QO%Q#e!R> #FD7$$\"1+++],s`KFD$\"1P*)o'=!H8BFD7$$\"1mm;zM)>L$FD$\"1-*Q,AJSU#FD7$$ \"1+++qfaHFD7$$\"1++]7y h]PFD$\"1m:4PizaIFD7$$\"1nmm')fdLQFD$\"1nT\\Ts%p=$FD7$$\"1nm;WV*f(QFD$ \"1/4t%F* 7$F6$\"1^3YM\"eP_%F*7$F9$\"1CM)*G&>#)y%F*7$F?$\"19:+)p1u*\\F*7$Fjo$\"1 hOSf:?(>&F*7$Fdp$\"1xFH6gu.aF*7$Fip$\"1JW[UO_CcF*7$F^q$\"1qZaq!)RzeF*7 $Fcq$\"14WjtFiMhF*7$Fhq$\"1t*\\;D/BY'F*7$F]r$\"1.%H0y]&QoF*7$Fbr$\"1.t ]!\\M%[sF*7$Fgr$\"1oY4)\\`>m(F*7$F\\s$\"1Zr$Q8s_?)F*7$Fas$\"1p'e)))*4# 4()F*7$Ffs$\"1LiEd]5X$*F*7$F[t$\"1=NL'>u;&**F*7$F`t$\"1'3*y1fJm5FD7$Fe t$\"1FhfD#4%Q6FD7$Fjt$\"1`S[N@+=7FD7$F_u$\"1q.!4A&*[H\"FD7$Fdu$\"15')G 3Wx\"Q\"FD7$Fiu$\"1*3N)=C>w9FD7$F^v$\"1h&HILBFD7$Fgw$\" 1#es_Ks[3#FD7$F\\x$\"1zsYXk!R>#FD7$Fax$\"1r;o#z!H8BFD7$Ffx$\"1Sg)y#>.C CFD7$F[y$\"1(4gL%zzZDFD7$F`y$\"1[)RB8Xnm#FD7$Fey$\"1#R]TO$e$z#FD7$Fjy$ \"1;`3;=**>HFD7$F_z$\"18'zK^(zaIFD7$Fdz$\"1<4:L([p=$FD7$F^[l$\"18DiboP CLFD7$F`^l$\"1*3d\"4_)GY$FD7$Ff^l$\"1+cMaf0#f$FD7$Fi^l$\"1hKN^ZFUPFD7$ F\\_l$\"1C&QbQ]&yQFD7$F__l$\"1q0I*=$zDSFD7$Fb_l$\"1eU&y&yboTFD7$Fe_l$ \"1(z??yHRK%FD-Fh_l6&Fj_lF[`lF+F+-F_`l6#%&POINTG-%+AXESLABELSG6$Q\"x6 \"%!G-%%VIEWG6$;$\"\"&!\"\"$\"#XFijl%(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 180 "In the above plot, we see that the p iecewise function representing the computed numerical solution follows the exact solution closely on the interval where it was computed, nam ely " }{XPPEDIT 18 0 "1 .. 4;" "6#;\"\"\"\"\"%" }{TEXT -1 2 " ." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 47 "Exploiting the Hybrid Symbolic-Numeric Paradigm" }}{PARA 0 "" 0 "" {TEXT -1 46 "See the paper (available as a postscript file)" }} {PARA 0 "" 0 "" {TEXT -1 17 " K.O. Geddes, " }{TEXT 295 80 "Generat ing Numerical ODE Formulas via a Symbolic Calculus of Divided Differen ces" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "referenced in the w eb page " }{TEXT 296 47 "http://daisy.uwaterloo.ca/~kogeddes/papers.ht ml" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "References" }}{PARA 0 "" 0 "" {TEXT -1 135 "[Bl uman89] G.W. Bluman and S. Kumei, Symmetries and Differential Equatio ns. Applied Mathematical Sciences, 81, Springer-Verlag, 1989." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "[ChebTer rab97] E.S. Cheb-Terrab, L.G.S. Duarte, and L.A.C.P. da Mota. Comput er Physics Communciations, 101 (1997), pp. 254." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 171 "[Corless96] R.M. Corles s, G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey and D.E. Knuth, On the Lambe rt W Function. Advances in Computational Mathematics, 5 (1996), pp. 3 29-359." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 147 "[Geddes92] K.O. Geddes and G.J. Fee, Hybrid symbolic-numeric int egration in Maple. Proceedings of ISSAC'92, ACM Press, New York, 1992 , pp. 36-41." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 223 "[Geddes95] K.O. Geddes and G. Labahn, Symbolic and numeric in tegration in Maple. Proceedings of The First Asian Technology Confere nce in Mathematics, The Association of Mathematics Educators, Singapor e, 1995, pp. 377-386." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 264 "[Kahan93] W. Kahan, Unconventional numerical methods for trajectory calculations. Manuscript, prepared for a workshop on \+ Integration Algorithms for Classical Mechanics, Fields Institute for R esearch in Mathematical Sciences, Waterloo, Ontario, Canada, Oct. 1993 ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "[Ka mke59] E. Kamke, Differentialgleichungen: Losungsmethoden und Losunge n. Chelsea Publishing Co, New York, 1959." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 109 "[Murphy60] G.M. Murphy, Ordinary Differential Equations and their solutions. Van Nostrand, Princeton, 1960." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 161 "[Stephani89] H. Stephani, Differential equations: their solutio n using symmetries, ed. M.A.H. MacCallum, Cambridge University Press, \+ New York and London, 1989." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "[Zwillinger92] D. Zwillinger, Handbook of Diffe rential Equations, 2nd edition, Academic Press, 1992." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "1 0 0" 8 }{VIEWOPTS 1 1 0 1 1 1803 }