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} {PSTYLE "" 0 271 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 272 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 273 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT 256 56 "Method of Undetermined \+ Coefficients - Annihilator Method" }}{PARA 19 "" 0 "" {TEXT 258 11 "2. 5epR4.mws" }}{PARA 259 "" 0 "" {TEXT 257 14 "Brian E. Blank" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 84 "Click on a [+] sign to expand a section. Click on a [-] sign to collapse a section." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 69 "Characteristic Equa tion of Constant Coefficient Homogeneous Equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "The following home-brewe d function converts a differential equation into its characteristic eq uation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 782 "ode2charEqn := proc()\n local r, jj, yy, xx, poly, \+ List, eqn_order;\n global _r;\n if nargs < 2 or nargs > 3 then\n ER ROR(`ode2charEqn expects two or three arguments`);\n elif not type(ar gs[1],equation) then\n ERROR(`ode2charEqn expects its first argument \+ to be a differential equation.`):\n elif not type(args[2], function) \+ then\n ERROR(`ode2charEqn expects its second argument to be a differe ntial equation.`):\n elif nargs=3 and not type(args[3], name) then\n \+ ERROR(`ode2charEqn expects its first argument to be a differential eq uation.`):\n elif nargs = 3 then r := args[3];\n else r := _r;\n fi ;\n yy := op(0,args[2]);\n xx := op(1,args[2]);\n List := DEtools[c onvertAlg](args[1],args[2]);\n poly := sum(List[1][jj]*r^(jj-1),jj=1. .nops(List[1]));\n RETURN(poly=0);\n end;\n \n " }}{PARA 12 "" 1 " " {XPPMATH 20 "6#>%,ode2charEqnG:6\"6)%\"rG%#jjG%#yyG%#xxG%%polyG%%Lis tG%*eqn_orderGF&F&C(@-529#\"\"#2\"\"$F3-%&ERRORG6#%Kode2charEqn~expect s~two~or~three~argumentsG4-%%typeG6$&9\"6#\"\"\"%)equationG-F86#%`oode 2charEqn~expects~its~first~argument~to~be~a~differential~equation.G4-F =6$&F@6#F4%)functionG-F86#%aoode2charEqn~expects~its~second~argument~t o~be~a~differential~equation.G3/F3F64-F=6$&F@6#F6%%nameGFDFQ>8$FU>FY%# _rG>8&-%#opG6$\"\"!FJ>8'-Fin6$FBFJ>8)-&%(DEtoolsG6#%+convertAlgG6$F?FJ >8(-%$sumG6$*&&&FaoFA6#8%FB)FY,&FapFB!\"\"FBFB/Fap;FB-%%nopsG6#F_p-%'R ETURNG6#/FioF[oF&6#Fen" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "A typical call would be " }{MPLTEXT 1 0 42 "ode2cha rEqn(ode_involving_y(x) , y(x) , r)" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 310 8 "Example:" }{TEXT 314 38 " \nFind the characteristic equation of " }{TEXT 311 4 " " } {XPPEDIT 312 1 "diff(y(x),x,x,x,x,x)+7*diff(y(x),x,x,x,x)+diff(y(x),x) -Pi*y(x) = 2/(1+x^2)" "/,*-%%diffG6(-%\"yG6#%\"xGF*F*F*F*F*\"\"\"*&\" \"(F+-F%6'-F(6#F*F*F*F*F*F+F+-F%6$-F(6#F*F*F+*&%#PiGF+-F(6#F*F+!\"\"*& \"\"#F+,&F+F+*$F*F " 0 "" {MPLTEXT 1 0 72 "ode := diff(y(x),x$5)+7*diff(y(x),x$4)+diff(y(x),x)-Pi*y(x) = 2/(1 +x^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG/,*-%%diffG6$-F(6$-F( 6$-F(6$-F(6$-%\"yG6#%\"xGF5F5F5F5F5\"\"\"F*\"\"(F0F6*&%#PiGF6F2F6!\"\" ,$*$,&F6F6*$F5\"\"#F6F:F?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "ode2charEqn(ode , y(x) , r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/, *%#PiG!\"\"%\"rG\"\"\"*$F'\"\"%\"\"(*$F'\"\"&F(\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 315 123 "Remember: Any home-m ade function must be loaded before use. In this worksheet that means e xecuting the code of the function" }{TEXT -1 2 " " }{MPLTEXT 1 0 11 " ode2charEqn" }{TEXT -1 2 " ." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 35 " Method of Undetermined Coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1:" }}{PARA 0 "" 0 "" {TEXT -1 33 "Solve the nonhomogeneous equation" }{TEXT 316 2 " " } {XPPEDIT 317 1 "diff(y(x),x,x) - 3*diff(y(x),x)+2*y(x)=5*exp(3*x)-7*x^ 2*exp(7*x)" "/,(-%%diffG6%-%\"yG6#%\"xGF*F*\"\"\"*&\"\"$F+-F%6$-F(6#F* F*F+!\"\"*&\"\"#F+-F(6#F*F+F+,&*&\"\"&F+-%$expG6#*&F-F+F*F+F+F+*(\"\"( F+*$F*F4F+-F;6#*&F?F+F*F+F+F2" }}{PARA 0 "" 0 "" {TEXT 318 9 "Solution :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "nonhomog := diff(y(x),x,x)-3 *diff(y(x),x)+2*y(x) = 5*exp(3*x)-7*x^2*exp(3*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)nonhomogG/,(-%%diffG6$-F(6$-%\"yG6#%\"xGF/F/\"\"\"F* !\"$F,\"\"#,&-%$expG6#,$F/\"\"$\"\"&*&F/F2F4F0!\"(" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "The set of linear combi nations of members of " }{TEXT 319 1 " " }{XPPEDIT 320 1 "rhsSet = \{e xp(3*x),x*exp(3*x),x^2*exp(3*x)\}" "/%'rhsSetG<%-%$expG6#*&\"\"$\"\"\" %\"xGF**&F+F*-F&6#*&F)F*F+F*F**&F+\"\"#-F&6#*&F)F*F+F*F*" }{TEXT -1 146 " is closed under differentiation. Let us make sure that none o f these functions appears in the solution of the associated homogeneou s equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "assoc_homog := lhs (nonhomog) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,assoc_homogG/,(- %%diffG6$-F(6$-%\"yG6#%\"xGF/F/\"\"\"F*!\"$F,\"\"#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "ode2charEqn(assoc_homog, y(x) , r); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(\"\"#\"\"\"%\"rG!\"$*$F'F%F&\" \"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "solve(\", r);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "assoc_homog_soln := y(x) = c[1]*exp(x) + c[2]*exp( 2*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%1assoc_homog_solnG/-%\"yG6# %\"xG,&*&&%\"cG6#\"\"\"F/-%$expGF(F/F/*&&F-6#\"\"#F/-F16#,$F)F5F/F/" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "Since t hese two summands do not appear in" }{MPLTEXT 1 0 7 " rhsSet" }{TEXT -1 85 ", we may seek a particular solution that is a linear combinati on of the elements of " }{MPLTEXT 1 0 6 "rhsSet" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "particular_soln := y(x) = C1 *exp(3*x) + C2*x*exp(3*x) + C3*x^2*exp(3*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0particular_solnG/-%\"yG6#%\"xG,(*&%#C1G\"\"\"-%$expG 6#,$F)\"\"$F-F-*(%#C2GF-F)F-F.F-F-*(%#C3GF-F)\"\"#F.F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "subs(particular_soln , nonhomog);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,,-%%diffG6$-F&6$,(*&%#C1G\"\"\"-%$ expG6#,$%\"xG\"\"$F-F-*(%#C2GF-F2F-F.F-F-*(%#C3GF-F2\"\"#F.F-F-F2F2F-F (!\"$F+F8F4F8F6F8,&F.\"\"&*&F2F8F.F-!\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,.* &%#C1G\"\"\"-%$expG6#,$%\"xG\"\"$F'\"\"#*&%#C2GF'F(F'F-*(F0F'F,F'F(F'F .*&%#C3GF'F(F'F.*(F3F'F,F'F(F'\"\"'*(F3F'F,F.F(F'F.,&F(\"\"&*&F,F.F(F' !\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "solve(identity(\",x ), \{C1, C2, C3\} );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%#C3G#!\"( \"\"#/%#C2G#\"#@F(/%#C1G#!#R\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "particular_soln := subs(\", particular_soln);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%0particular_solnG/-%\"yG6#%\"xG,(-%$ expG6#,$F)\"\"$#!#R\"\"%*&F)\"\"\"F+F4#\"#@\"\"#*&F)F7F+F4#!\"(F7" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Now our g eneral solution is given by:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "ge neral_soln := y(x) = rhs(assoc_homog_soln) + rhs(particular_soln);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%-general_solnG/-%\"yG6#%\"xG,,*&&%\" cG6#\"\"\"F/-%$expGF(F/F/*&&F-6#\"\"#F/-F16#,$F)F5F/F/-F16#,$F)\"\"$#! #R\"\"%*&F)F/F9F/#\"#@F5*&F)F5F9F/#!\"(F5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Verification:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "subs(general_soln, nonhomog);" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#/,0-%%diffG6$-F&6$,,*&&%\"cG6#\"\"\"F/-%$expG6#%\"xGF /F/*&&F-6#\"\"#F/-F16#,$F3F7F/F/-F16#,$F3\"\"$#!#R\"\"%*&F3F/F;F/#\"#@ F7*&F3F7F;F/#!\"(F7F3F3F/F(!\"$F+F7F4F7F;#F@F7FBFDFEFG,&F;\"\"&FEFG" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%$expG6#,$%\"xG\"\"$\"\"&*&F)\"\"#F%\"\"\"! \"(F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "which is clearly true!" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "Example 2: \+ (Edwards and Penney page 157 #13.)" }}{PARA 0 "" 0 "" {TEXT -1 33 "So lve the nonhomogeneous equation" }{TEXT 321 2 " " }{XPPEDIT 322 1 "di ff(y(x),x,x) +2*diff(y(x),x)+5*y(x)=exp(x)*sin(x)" "/,(-%%diffG6%-%\"y G6#%\"xGF*F*\"\"\"*&\"\"#F+-F%6$-F(6#F*F*F+F+*&\"\"&F+-F(6#F*F+F+*&-%$ expG6#F*F+-%$sinG6#F*F+" }}{PARA 0 "" 0 "" {TEXT 323 9 "Solution:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "nonhomog := diff(y(x),x,x)+2 *diff(y(x),x)+2*y(x) = exp(x)*sin(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)nonhomogG/,(-%%diffG6$-F(6$-%\"yG6#%\"xGF/F/\"\"\"F*\"\"#F,\"\"& *&-%$expGF.F0-%$sinGF.F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "The set of linear combinations of members of " }{TEXT 324 1 " " }{XPPEDIT 325 1 "rhsSet = \{exp(x)*sin(x),exp(x )*cos(x)\}" "/%'rhsSetG<$*&-%$expG6#%\"xG\"\"\"-%$sinG6#F)F**&-F'6#F)F *-%$cosG6#F)F*" }{TEXT -1 146 " is closed under differentiation. Le t us make sure that none of these functions appears in the solution of the associated homogeneous equation:" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "assoc_homog := lhs(nonhomog) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,assoc_homogG/,(-%%diffG6$-F(6$-%\"yG6#%\"xGF/F/\"\" \"F*\"\"#F,\"\"&\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "od e2charEqn(assoc_homog, y(x) , r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ ,(\"\"&\"\"\"%\"rG\"\"#*$F'F(F&\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "solve(\", r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&! \"\"\"\"\"%\"IG\"\"#,&F$F%F&!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "assoc_homog_soln := y(x) = c[1]*exp(-x)*cos(2*x) + c[ 2]*exp(-x)*sin(2*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%1assoc_homog _solnG/-%\"yG6#%\"xG,&*(&%\"cG6#\"\"\"F/-%$expG6#,$F)!\"\"F/-%$cosG6#, $F)\"\"#F/F/*(&F-6#F9F/F0F/-%$sinGF7F/F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "Since these two summands do not ap pear in" }{MPLTEXT 1 0 7 " rhsSet" }{TEXT -1 85 ", we may seek a part icular solution that is a linear combination of the elements of " } {MPLTEXT 1 0 6 "rhsSet" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "particular_soln := y(x) = C1*exp(x)*sin(x) + C2*exp(x)*cos(x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0particular_solnG/-%\"yG6#%\"xG, &*(%#C1G\"\"\"-%$expGF(F--%$sinGF(F-F-*(%#C2GF-F.F--%$cosGF(F-F-" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "subs(particular_soln , nonho mog);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,*-%%diffG6$-F&6$,&*(%#C1G\" \"\"-%$expG6#%\"xGF--%$sinGF0F-F-*(%#C2GF-F.F--%$cosGF0F-F-F1F1F-F(\" \"#F+\"\"&F4F9*&F.F-F2F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,**(%#C1G\"\"\"-%$e xpG6#%\"xGF'-%$cosGF*F'\"\"%*(%#C2GF'F(F'-%$sinGF*F'!\"%*(F&F'F(F'F1F' \"\"(*(F0F'F(F'F,F'F5*&F(F'F1F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "solve(identity(\",x), \{C1, C2\} );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%#C2G#!\"%\"#l/%#C1G#\"\"(F(" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 44 "particular_soln := subs(\", particular_soln); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0particular_solnG/-%\"yG6#%\"xG, &*&-%$expGF(\"\"\"-%$sinGF(F.#\"\"(\"#l*&F,F.-%$cosGF(F.#!\"%F3" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Now our g eneral solution is given by:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "ge neral_soln := y(x) = rhs(assoc_homog_soln) + rhs(particular_soln);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%-general_solnG/-%\"yG6#%\"xG,**(&%\" cG6#\"\"\"F/-%$expG6#,$F)!\"\"F/-%$cosG6#,$F)\"\"#F/F/*(&F-6#F9F/F0F/- %$sinGF7F/F/*&-F1F(F/-F>F(F/#\"\"(\"#l*&F@F/-F6F(F/#!\"%FD" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Verification:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "subs(general_soln, nonhomog) ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,.-%%diffG6$-F&6$,**(&%\"cG6#\" \"\"F/-%$expG6#,$%\"xG!\"\"F/-%$cosG6#,$F4\"\"#F/F/*(&F-6#F:F/F0F/-%$s inGF8F/F/*&-F16#F4F/-F?FBF/#\"\"(\"#l*&FAF/-F7FBF/#!\"%FFF4F4F/F(F:F+ \"\"&F;FKF@#FE\"#8FG#FJFMF@" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%$expG6#%\"xG \"\"\"-%$sinGF'F)F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 22 "which is clearly true!" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 44 "Example 3: (Edwards and Penney page 156 #3.)" }}{PARA 0 "" 0 "" {TEXT 326 5 "Solve" }{TEXT 330 1 " " }{TEXT -1 2 " " }{TEXT 327 2 " \+ " }{XPPEDIT 328 1 "diff(diff(y(x),x),x)-diff(y(x),x)-6*y(x) = 2*sin(3* x)" "/,(-%%diffG6$-F%6$-%\"yG6#%\"xGF,F,\"\"\"-F%6$-F*6#F,F,!\"\"*&\" \"'F--F*6#F,F-F2*&\"\"#F--%$sinG6#*&\"\"$F-F,F-F-" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 329 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Let us first name the original equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 69 "non_homog_eqn := diff(y(x),x,x) - diff(y(x),x) - 6*y(x) = 2*sin(3*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.non_homo g_eqnG/,(-%%diffG6$-F(6$-%\"yG6#%\"xGF/F/\"\"\"F*!\"\"F,!\"',$-%$sinG6 #,$F/\"\"$\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "We must find the general solution to the associated homo geneous equation that we get by making the rhs zero:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "assoc_homog_eqn := lhs(non_homog_eqn) = 0;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%0assoc_homog_eqnG/,(-%%diffG6$-F(6$- %\"yG6#%\"xGF/F/\"\"\"F*!\"\"F,!\"'\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "We do this by forming the associat ed polynomial (replace each " }{TEXT 333 2 " " }{XPPEDIT 334 1 "d/(d* x)" "*&%\"dG\"\"\"*&F#F$%\"xGF$!\"\"" }{TEXT -1 22 " by a variable, \+ say " }{TEXT 331 2 " " }{XPPEDIT 332 1 "w" "I\"wG6\"" }{TEXT -1 16 ") and factoring:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 35 "ode2charEqn(assoc_homog_eqn, y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(!\"'\"\"\"%#_rG!\"\"*$F'\"\"#F&\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "solve(\");" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$!\"#\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "From this we know that the general solution of \+ the associated homogeneous equation is:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "assoc_homog_soln := y(x ) = c1*exp(-2*x)+c2*exp(3*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%1as soc_homog_solnG/-%\"yG6#%\"xG,&*&%#c1G\"\"\"-%$expG6#,$F)!\"#F-F-*&%#c 2GF--F/6#,$F)\"\"$F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "But we are careful so we verify before proceeding:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "subs(assoc_homog_soln,assoc_homog_eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,*-%%diffG6$-F&6$,&*&%#c1G\"\"\"-%$expG6#,$%\"xG!\"#F- F-*&%#c2GF--F/6#,$F2\"\"$F-F-F2F2F-F(!\"\"F+!\"'F4F;\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "simplify(lhs(\")-rhs(\"));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 63 "Next, we look at the right hand side \+ of the original equation:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "rhs(original_ode);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$sinG6#, $%\"xG\"\"$\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Since all derivatives of all orders of " }{XPPEDIT 337 1 "sin(3*x)" "-%$sinG6#*&\"\"$\"\"\"%\"xGF'" }{TEXT -1 28 " are linea r combinations of" }{TEXT 338 2 " " }{XPPEDIT 339 1 "sin(3*x)" "-%$si nG6#*&\"\"$\"\"\"%\"xGF'" }{TEXT -1 7 " and " }{TEXT 340 1 " " } {XPPEDIT 341 1 "cos(3*x)" "-%$cosG6#*&\"\"$\"\"\"%\"xGF'" }{TEXT -1 79 ", we know that the Method of Undetermined Coefficients is applica ble. We try " }{TEXT 335 2 " " }{XPPEDIT 336 1 "y(x) = A*sin(3*x)+B* cos(3*x)" "/-%\"yG6#%\"xG,&*&%\"AG\"\"\"-%$sinG6#*&\"\"$F*F&F*F*F**&% \"BGF*-%$cosG6#*&F/F*F&F*F*F*" }{TEXT -1 55 " as our particular solu tion of the original equation." }}{PARA 0 "" 0 "" {TEXT -1 99 "\nOnce \+ we solve for these undetermined coefficients, our solution to the orig inal requation will be:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "particu lar_soln := y(x) = A*sin(3*x)+B*cos(3*x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0particular_solnG/-%\"yG6#%\"xG,&*&%\"AG\"\"\"-%$sinG 6#,$F)\"\"$F-F-*&%\"BGF--%$cosGF0F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "non_homog_soln := y(x) = rhs(particular_soln) + rhs(a ssoc_homog_soln);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/non_homog_soln G/-%\"yG6#%\"xG,**&%\"AG\"\"\"-%$sinG6#,$F)\"\"$F-F-*&%\"BGF--%$cosGF0 F-F-*&%#c1GF--%$expG6#,$F)!\"#F-F-*&%#c2GF--F:F0F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 63 "\nNotice that not all constants are created equal. The constants" }{TEXT 345 2 " " }{XPPEDIT 346 1 "c1" "I#c1G6\"" }{TEXT -1 8 " and " }{TEXT 347 1 " " }{XPPEDIT 348 1 "c2" "I#c2G6\"" } {TEXT -1 141 " cannot be solved for unless other information, such a s an initial condition is given. We are about to solve for the other \+ two constants. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "subs(particu lar_soln, non_homog_eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,*-%%dif fG6$-F&6$,&*&%\"AG\"\"\"-%$sinG6#,$%\"xG\"\"$F-F-*&%\"BGF--%$cosGF0F-F -F2F2F-F(!\"\"F+!\"'F4F9,$F.\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "identity_to_determine_coefficients := simplify(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%Cidentity_to_determine_coefficient sG/,**&%\"AG\"\"\"-%$sinG6#,$%\"xG\"\"$F)!#:*&%\"BGF)-%$cosGF,F)F0*&F( F)F3F)!\"$*&F2F)F*F)F/,$F*\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 281 "Next we \+ solve for the undetermined coefficients. This amounts to grouping like terms on the left hand side and equating the resulting coefficients w ith their counterparts on the right hand side. This will give two equa tions. Some of this we can do in our head. The coefficient of " } {XPPEDIT 342 1 "sin(3*x)" "-%$sinG6#*&\"\"$\"\"\"%\"xGF'" }{TEXT -1 29 " on the left hand side is " }{XPPEDIT 344 0 "-15*A+3*B" ",&*&\" #:\"\"\"%\"AGF%!\"\"*&\"\"$F%%\"BGF%F%" }{TEXT -1 27 " and the coeff icient of " }{XPPEDIT 343 1 "cos(3*x)" "-%$cosG6#*&\"\"$\"\"\"%\"xGF' " }{TEXT -1 7 " is " }{XPPEDIT 19 1 "-3*A-15*B" ",&*&\"\"$\"\"\"%\" AGF%!\"\"*&\"#:F%%\"BGF%F'" }{TEXT -1 41 ". These correspond to the c oefficients " }{XPPEDIT 19 1 "2" "\"\"#" }{TEXT -1 5 "and " } {XPPEDIT 19 1 "0" "\"\"!" }{TEXT -1 59 " on the right hand side. Thi s gives the set of equations:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "coeff_eqn_set := \{-15*A+3*B = 2, -3*A-15*B = 0\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.coeff_eqn _setG<$/,&%\"AG!\"$%\"BG!#:\"\"!/,&F(F+F*\"\"$\"\"#" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "We solve this set of eq uations for the set of unknown coefficients:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "determined_coefficients := solve(coeff_eqn_set, \{A,B \} );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%8determined_coefficientsG<$ /%\"AG#!\"&\"#R/%\"BG#\"\"\"F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "We might \+ also have proceeded as follows to solve for these coefficients. We ha ve used the name " }{MPLTEXT 1 0 34 "identity_to_determine_coefficient s" }{TEXT -1 33 " to suggest an identity but to " }{TEXT 349 6 " MAP LE" }{TEXT -1 37 " we may as well have used the name " }{MPLTEXT 1 0 21 "TheGrandOldDukeOfYork" }{TEXT -1 59 ". (You know, the guy with 10000 men.) In order to let " }{TEXT 350 6 " MAPLE" }{TEXT -1 51 " know that this name represents an identity in " }{MPLTEXT 1 0 1 " x" }{TEXT -1 26 " we have to write it as " }{MPLTEXT 1 0 48 "identit y( identity_to_determine_coefficients, x)" }{TEXT -1 52 " before sol ving for the undetermined coefficients." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "solve(identity( identit y_to_determine_coefficients, x), \{A,B\} );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"AG#!\"&\"#R/%\"BG#\"\"\"F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "subs(determined_coefficients, non_homog_soln);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,*-%$sinG6#,$F'\"\"$#!\" &\"#R-%$cosGF+#\"\"\"F0*&%#c1GF4-%$expG6#,$F'!\"#F4F4*&%#c2GF4-F8F+F4F 4" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "And our check:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "subs(\", non_homog _eqn);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,.-%%diffG6$-F&6$,*-%$sinG6 #,$%\"xG\"\"$#!\"&\"#R-%$cosGF-#\"\"\"F3*&%#c1GF7-%$expG6#,$F/!\"#F7F7 *&%#c2GF7-F;F-F7F7F/F/F7F(!\"\"F+#\"#5\"#8F4#F>FEF8!\"'F?FG,$F+\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "simplify(lhs(\")-rhs(\")) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "Success!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 18 "Annihilator Method" }}{PARA 0 "" 0 "" {TEXT -1 360 "In the Method of Undetermined Coefficients, one looks to see whether the right hand side belongs to a finite set of functions whose linear com binations are closed under differentiation. If one of these functions \+ appears in the solution of the associated homogeneous equation, then t he application of the Method of Undetermined Coefficients is more comp licated." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "In this section we will consider an alternative approach: the Anni hilator Method. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The idea is simple. Given the differential equation " } {TEXT 352 2 " " }{XPPEDIT 353 1 "d^n*y(x)/(d*x^n) + a[n-1]*d^(n-1)*y( x)/(d*x^(n-1))+`...`+a[0]*y(x)=f(x)" "/,**()%\"dG%\"nG\"\"\"-%\"yG6#% \"xGF(*&F&F()F,F'F(!\"\"F(**&%\"aG6#,&F'F(F(F/F()F&,&F'F(F(F/F(-F*6#F, F(*&F&F()F,,&F'F(F(F/F(F/F(%$...GF(*&&F26#\"\"!F(-F*6#F,F(F(-%\"fG6#F, " }{TEXT -1 492 ", find a differential operator that annihilates the \+ right side. If you apply that operator to the original equation, then \+ a homogeneous equation results. The solution to that \"annihilated hom ogeneous equation\" can then be substituted into the original nonhomog eneous equation in order to determine particular values of some of the constants. The examples will illustrate better than a lot of chit-cha t. The important thing is to be able to identify the annihilator. Here are common examples:" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 17 " " }{TEXT 354 10 "Expression" } {TEXT -1 55 " " }{TEXT 355 12 " Annihilator" }}{PARA 0 "" 0 "" {TEXT -1 4 " " } {TEXT 356 2 " " }}{PARA 268 "" 0 "" {TEXT -1 20 " \+ " }{XPPEDIT 19 1 "x^n" ")%\"xG%\"nG" }{TEXT -1 45 " \+ " }{XPPEDIT 19 1 "z ->diff(z,x)@@(n+1)" ":6# %\"zG7\"6$%)operatorG%&arrowG6\"-%#@@G6$-%%diffG6$F$%\"xG,&%\"nG\"\"\" F3F3F)F)" }}{PARA 269 "" 0 "" {TEXT -1 16 " " } {XPPEDIT 19 1 "x^n*exp(k*x)" "*&)%\"xG%\"nG\"\"\"-%$expG6#*&%\"kGF&F$F &F&" }{TEXT -1 38 " " }{XPPEDIT 19 1 "z ->(diff(z,x)-k*z)@@(n+1)" ":6#%\"zG7\"6$%)operatorG%&arrowG6\" -%#@@G6$,&-%%diffG6$F$%\"xG\"\"\"*&%\"kGF2F$F2!\"\",&%\"nGF2F2F2F)F)" }}{PARA 270 "" 0 "" {TEXT -1 8 " " }}{PARA 271 "" 0 "" {TEXT -1 12 " " }{XPPEDIT 19 1 "A*cos(omega*x)+B*sin(omega*x)" ", &*&%\"AG\"\"\"-%$cosG6#*&%&omegaGF%%\"xGF%F%F%*&%\"BGF%-%$sinG6#*&F*F% F+F%F%F%" }{TEXT -1 19 " " }{XPPEDIT 19 1 "z -> (dif f(z,x,x)+omega^2*z" ":6#%\"zG7\"6$%)operatorG%&arrowG6\",&-%%diffG6%F$ %\"xGF.\"\"\"*&%&omegaG\"\"#F$F/F/F)F)" }{TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{TEXT 357 8 " " }{XPPEDIT 358 1 "x^n* (A*cos(omega*x)+B*sin(omega*x))" "*&)%\"xG%\"nG\"\"\",&*&%\"AGF&-%$cos G6#*&%&omegaGF&F$F&F&F&*&%\"BGF&-%$sinG6#*&F.F&F$F&F&F&F&" }{TEXT 359 12 " " }{XPPEDIT 360 1 "z -> ((diff(z,x,x)+omega^2*z)@@(n+1 )" ":6#%\"zG7\"6$%)operatorG%&arrowG6\"-%#@@G6$,&-%%diffG6%F$%\"xGF1\" \"\"*&%&omegaG\"\"#F$F2F2,&%\"nGF2F2F2F)F)" }{TEXT 361 4 " " }} {PARA 272 "" 0 "" {TEXT -1 8 " " }{XPPEDIT 19 1 "exp(r*x)*(A*co s(omega*x)+B*sin(omega*x))" "*&-%$expG6#*&%\"rG\"\"\"%\"xGF(F(,&*&%\"A GF(-%$cosG6#*&%&omegaGF(F)F(F(F(*&%\"BGF(-%$sinG6#*&F1F(F)F(F(F(F(" } {TEXT -1 9 " " }{XPPEDIT 19 1 "z -> (diff(z,x,x)-2*r*diff(z,x) +(r^2+omega^2)*z" ":6#%\"zG7\"6$%)operatorG%&arrowG6\",(-%%diffG6%F$% \"xGF.\"\"\"*(\"\"#F/%\"rGF/-F,6$F$F.F/!\"\"*&,&*$F2F1F/*$%&omegaGF1F/ F/F$F/F/F)F)" }{TEXT -1 8 " " }}{PARA 273 "" 0 "" {TEXT -1 5 " \+ " }{XPPEDIT 19 1 "x^n*exp(r*x)*(A*cos(omega*x)+B*sin(omega*x))" "* ()%\"xG%\"nG\"\"\"-%$expG6#*&%\"rGF&F$F&F&,&*&%\"AGF&-%$cosG6#*&%&omeg aGF&F$F&F&F&*&%\"BGF&-%$sinG6#*&F3F&F$F&F&F&F&" }{TEXT -1 7 " " }{XPPEDIT 19 1 "z -> ((diff(z,x,x)-2*r*diff(z,x)+(r^2+omega^2)*z)@@(n+ 1)" ":6#%\"zG7\"6$%)operatorG%&arrowG6\"-%#@@G6$,(-%%diffG6%F$%\"xGF1 \"\"\"*(\"\"#F2%\"rGF2-F/6$F$F1F2!\"\"*&,&*$F5F4F2*$%&omegaGF4F2F2F$F2 F2,&%\"nGF2F2F2F)F)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 38 "An Example from the Suggested Problems" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "P age 156 #9." }}{PARA 0 "" 0 "" {TEXT 292 5 "Solve" }{TEXT 296 1 " " } {TEXT -1 2 " " }{TEXT 293 2 " " }{XPPEDIT 294 1 "diff(diff(y(x),x),x )+2*diff(y(x),x)-3*y(x) = 1+x*exp(x)" "/,(-%%diffG6$-F%6$-%\"yG6#%\"xG F,F,\"\"\"*&\"\"#F--F%6$-F*6#F,F,F-F-*&\"\"$F--F*6#F,F-!\"\",&F-F-*&F, F--%$expG6#F,F-F-" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 295 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Let us first name th e original equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "non_homog_ eqn := diff(y(x),x$2)+2*diff(y(x),x)-3*y(x) = 1+x*exp(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.non_homog_eqnG/,(-%%diffG6$-F(6$-%\"yG6#%\"x GF/F/\"\"\"F*\"\"#F,!\"$,&F0F0*&F/F0-%$expGF.F0F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "We must find the general solution to the associated homogeneous equation that we get by making the rhs zero:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "assoc_homog_ eqn := lhs(non_homog_eqn) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0a ssoc_homog_eqnG/,(-%%diffG6$-F(6$-%\"yG6#%\"xGF/F/\"\"\"F*\"\"#F,!\"$ \"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "We do this by forming the associated polynomial (replace each " } {TEXT 299 2 " " }{XPPEDIT 300 1 "d/(d*x)" "*&%\"dG\"\"\"*&F#F$%\"xGF$ !\"\"" }{TEXT -1 22 " by a variable, say " }{TEXT 297 2 " " } {XPPEDIT 298 1 "w" "I\"wG6\"" }{TEXT -1 16 ") and factoring:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "od e2charEqn(non_homog_eqn, y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,( !\"$\"\"\"%#_rG\"\"#*$F'F(F&\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "solve(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$!\"$\"\" \"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Fr om this we know that the general solution of the associated homogeneou s equation is:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "assoc_homog_soln := y(x) = c1*exp(-3*x)+c2*exp(x );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%1assoc_homog_solnG/-%\"yG6#%\" xG,&*&%#c1G\"\"\"-%$expG6#,$F)!\"$F-F-*&%#c2GF--F/F(F-F-" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "But we are careful so we verify before proceeding:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "subs(assoc_homog_soln, assoc _homog_eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,*-%%diffG6$-F&6$,&*& %#c1G\"\"\"-%$expG6#,$%\"xG!\"$F-F-*&%#c2GF--F/6#F2F-F-F2F2F-F(\"\"#F+ F3F4F3\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "simplify(lhs (\")-rhs(\"));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Next, we look at t he right hand side of the original equation:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "rhs(non_homog_eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$*&%\"xGF$-%$expG6#F&F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Since all derivatives of \+ all orders of " }{XPPEDIT 301 1 "1+x*exp(x)" ",&\"\"\"F#*&%\"xGF#-%$e xpG6#F%F#F#" }{TEXT -1 28 " are linear combinations of" }{TEXT 302 2 " " }{XPPEDIT 303 1 "1" "\"\"\"" }{TEXT -1 5 " , " }{XPPEDIT 305 1 "exp(x)" "-%$expG6#%\"xG" }{TEXT -1 9 " , and " }{XPPEDIT 304 1 "x*e xp(x)" "*&%\"xG\"\"\"-%$expG6#F#F$" }{TEXT -1 297 " , we know that th e Method of Undetermined Coefficients is applicable. Since one of thes e terms is already part of the solution of the associated homogeneous \+ equation, the Method of Undetermined coeffixcients is in the tricky ca se. We continue with the Method but taking the Annihilator approach." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 289 "The fi rst step is to look again at the right side. We have to figure out wha t differential operator \"annihilates\" it. (Otherwise put, we are doi ng backward reasoning to figure out a differential equation the right \+ hand side satisfies.) If we differentiate we annihilate the 1. If we \+ then " }}{PARA 0 "" 0 "" {TEXT -1 8 "apply " }{XPPEDIT 19 1 "(D-1)^2 " "*$,&%\"DG\"\"\"F%!\"\"\"\"#" }{TEXT -1 21 " we annihilate the " } {TEXT 306 1 " " }{XPPEDIT 307 1 "x*exp(x)" "*&%\"xG\"\"\"-%$expG6#F#F$ " }{TEXT -1 32 ". We must also annihilate the " }{XPPEDIT 351 1 "1" "\"\"\"" }{TEXT -1 51 " . That is done by the differentiation operato r " }{MPLTEXT 1 0 12 "z->diff(z,x)" }{TEXT -1 1 ":" }}{PARA 0 "" 0 " " {TEXT -1 10 "Therefore:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "annihilator := (z -> diff(z,x))@(z ->diff(z,x)-z)@@2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,annihilatorG- %\"@G6$:6#%\"zG6\"6$%)operatorG%&arrowGF+-%%diffG6$9$%\"xGF+F+-%#@@G6$ :F)F+F,F+,&F/\"\"\"F2!\"\"F+F+\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "annihilated_homog_eqn := annihilator(non_homog_eqn); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%6annihilated_homog_eqnG/,*-%%dif fG6$-F(6$-F(6$-F(6$-F(6$-%\"yG6#%\"xGF5F5F5F5F5\"\"\"F,!\"'F.\"\")F0! \"$\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "If the right hand side was not zero, or did not simplify to zero, that would be a sign of a misstep." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 34 "We solve the annihilated equation:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "annihilated_homog_soln := dsolve(annihilated_homog_eqn, y(x));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%7annihilated_homog_solnG/-%\"yG6#%\" xG,,%$_C1G\"\"\"*&%$_C2GF,-%$expGF(F,F,*&%$_C3GF,-F06#,$F)!\"$F,F,*(%$ _C4GF,F)F,F/F,F,*(%$_C5GF,F/F,F)\"\"#F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "The coefficients of the terms that already appear in " }{MPLTEXT 1 0 17 "assoc_homog_soln " }{TEXT -1 87 " cannot be solved for. If we were solving this by hand we would n ot bother at all with" }{TEXT 308 3 " " }{XPPEDIT 309 1 "_C2*exp(x)+ _C3*exp(-3*x)" ",&*&%$_C2G\"\"\"-%$expG6#%\"xGF%F%*&%$_C3GF%-F'6#,$*& \"\"$F%F)F%!\"\"F%F%" }{TEXT -1 231 ". (They will just disappear from our next calculations.) However, when using Maple, it is eaier just t o keep them along for the ride until they get bumped. We are really us ing the Method of Undetermined coefficients so we plug " }{MPLTEXT 1 0 22 "annihilated_homog_soln" }{TEXT -1 69 " into the original ode \+ and solve for the undetermined coefficients. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "subs(annihilated_homog_soln , non_homog_eqn);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/,0-%%diffG6$-F&6$,,%$_C1G\"\"\"*&%$_C 2GF,-%$expG6#%\"xGF,F,*&%$_C3GF,-F06#,$F2!\"$F,F,*(%$_C4GF,F2F,F/F,F,* (%$_C5GF,F/F,F2\"\"#F,F2F2F,F(F=F+F8F-F8F3F8F9F8F;F8,&F,F,*&F2F,F/F,F, " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "identity_to_determine_c oefficients := simplify(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%Cide ntity_to_determine_coefficientsG/,**&%$_C4G\"\"\"-%$expG6#%\"xGF)\"\"% *(%$_C5GF)F*F)F-F)\"\")*&F0F)F*F)\"\"#%$_C1G!\"$,&F)F)*&F-F)F*F)F)" }} }{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 314 "Next we \+ solve for the undetermined coefficients. This amounts to grouping like terms on the left hand side and equating the resulting coefficients w ith their counterparts on the right hand side. This will give three eq uations in three unknowns. Some of this we can do in our head, arrivin g at the set of equations:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "coeff_eqn_set := \{-3*_C1 = 1, 4*_C 4+2*_C5 = 0, 8*_C5 = 1\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.coeff_ eqn_setG<%/,$%$_C1G!\"$\"\"\"/,&%$_C4G\"\"%%$_C5G\"\"#\"\"!/,$F/\"\")F *" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "We \+ solve this set of equations for the set of unknown coefficients:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "determined_coefficients := s olve(coeff_eqn_set, \{_C1, _C4, _C5\} );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%8determined_coefficientsG<%/%$_C5G#\"\"\"\"\")/%$_C1G#!\"\"\" \"$/%$_C4G#F.\"#;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "Alternatively, we could h ave asked Maple to solve the identity in one swoop:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "solve(iden tity(identity_to_determine_coefficients, x), \{_C1, _C4, _C5\} );" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%$_C5G#\"\"\"\"\")/%$_C1G#!\"\"\"\" $/%$_C4G#F,\"#;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Either way:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "subs( determined_coefficients, annihilated_homog_soln);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,,#!\"\"\"\"$\"\"\"*&%$_C2GF,-%$expGF&F,F ,*&%$_C3GF,-F06#,$F'!\"$F,F,*&F'F,F/F,#F*\"#;*&F/F,F'\"\"##F,\"\")" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "And our \+ check:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "subs(\", non_homog_eqn); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,0-%%diffG6$-F&6$,,#!\"\"\"\"$\" \"\"*&%$_C2GF.-%$expG6#%\"xGF.F.*&%$_C3GF.-F26#,$F4!\"$F.F.*&F4F.F1F.# F,\"#;*&F1F.F4\"\"##F.\"\")F4F4F.F(F?F.F.F/F:F5F:F;#F-F=F>#F:FA,&F.F.F ;F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "simplify(lhs(\")-rhs (\"));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "Success!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 19 "Additional Examples" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{TEXT 259 4 " " } {XPPEDIT 260 1 "diff(y(x),x,x)+4*y(x)=cos(2*x)" "/,&-%%diffG6%-%\"yG6# %\"xGF*F*\"\"\"*&\"\"%F+-F(6#F*F+F+-%$cosG6#*&\"\"#F+F*F+" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "non_homog_eqn := diff(y(x),x,x)+4*y(x) = co s(2*x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.non_homog_eqnG/,&-%%dif fG6$-F(6$-%\"yG6#%\"xGF/F/\"\"\"F,\"\"%-%$cosG6#,$F/\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "annihilated_homog_eqn := map(z -> d iff(z,x$2) + 4*z, non_homog_eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %6annihilated_homog_eqnG/,(-%%diffG6$-F(6$-F(6$-F(6$-%\"yG6#%\"xGF3F3F 3F3\"\"\"F,\"\")F0\"#;\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "annihilated_homog_soln := dsolve( \" , y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%7annihilated_homog_solnG/-%\"yG6#%\"xG,**&%$_C1G\"\" \"-%$cosG6#,$F)\"\"#F-F-*&%$_C2GF--%$sinGF0F-F-*(%$_C3GF-F.F-F)F-F-*(% $_C4GF-F5F-F)F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "subs( \", non_homog_eqn);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,,-%%diffG6$-F &6$,**&%$_C1G\"\"\"-%$cosG6#,$%\"xG\"\"#F-F-*&%$_C2GF--%$sinGF0F-F-*(% $_C3GF-F.F-F2F-F-*(%$_C4GF-F6F-F2F-F-F2F2F-F+\"\"%F4F " 0 "" {MPLTEXT 1 0 12 "simplify(\");" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/,&*&%$_C3G\"\"\"-%$sinG6#,$%\"xG\"\"#F'!\"%*&%$ _C4GF'-%$cosGF*F'\"\"%F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "solve(identity(\",x),\{_C3,_C4\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #<$/%$_C3G\"\"!/%$_C4G#\"\"\"\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "non_homog_soln := subs(\", annihilated_homog_soln);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/non_homog_solnG/-%\"yG6#%\"xG,(*& %$_C1G\"\"\"-%$cosG6#,$F)\"\"#F-F-*&%$_C2GF--%$sinGF0F-F-*&F5F-F)F-#F- \"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Verification:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "subs(non_homog_soln, non_homog_eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,*-%%diffG6$-F&6$,(*&%$_C1G\"\"\"-%$cosG6#, $%\"xG\"\"#F-F-*&%$_C2GF--%$sinGF0F-F-*&F6F-F2F-#F-\"\"%F2F2F-F+F:F4F: F8F-F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "simplify(lhs(\")- rhs(\"));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT 261 4 " " }{XPPEDIT 262 1 "diff(y(x),x,x,x)-3*diff(y(x),x,x)+4*y(x)=exp(3*x)" "/,(-%%diffG6&-% \"yG6#%\"xGF*F*F*\"\"\"*&\"\"$F+-F%6%-F(6#F*F*F*F+!\"\"*&\"\"%F+-F(6#F *F+F+-%$expG6#*&F-F+F*F+" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "non_homog_eqn := diff(y(x),x,x,x)-3*diff(y( x),x,x)+4*y(x) = exp(3*x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.non_ homog_eqnG/,(-%%diffG6$-F(6$-F(6$-%\"yG6#%\"xGF1F1F1\"\"\"F*!\"$F.\"\" %-%$expG6#,$F1\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "anni hilated_homog_eqn := map(z -> diff(z,x) - 3*z, non_homog_eqn);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%6annihilated_homog_eqnG/,,-%%diffG6$ -F(6$-F(6$-F(6$-%\"yG6#%\"xGF3F3F3F3\"\"\"F*!\"'F.\"\"%F,\"\"*F0!#7\" \"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "annihilated_homog_so ln := dsolve( \" , y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%7annihi lated_homog_solnG/-%\"yG6#%\"xG,**&%$_C1G\"\"\"-%$expG6#,$F)\"\"$F-F-* &%$_C2GF--F/6#,$F)\"\"#F-F-*&%$_C3GF--F/6#,$F)!\"\"F-F-*(%$_C4GF-F5F-F )F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "subs(\", non_homog _eqn);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,.-%%diffG6$-F&6$-F&6$,**&% $_C1G\"\"\"-%$expG6#,$%\"xG\"\"$F/F/*&%$_C2GF/-F16#,$F4\"\"#F/F/*&%$_C 3GF/-F16#,$F4!\"\"F/F/*(%$_C4GF/F8F/F4F/F/F4F4F4F/F(!\"$F-\"\"%F6FEF " 0 "" {MPLTEXT 1 0 12 "simplify(\");" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&%$_C1G\"\"\"-%$expG6#,$%\"xG\"\" $F'\"\"%F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "solve(identit y(\",x),\{_C1\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#/%$_C1G#\"\"\" \"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "non_homog_soln := \+ subs(\", annihilated_homog_soln);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %/non_homog_solnG/-%\"yG6#%\"xG,*-%$expG6#,$F)\"\"$#\"\"\"\"\"%*&%$_C2 GF1-F,6#,$F)\"\"#F1F1*&%$_C3GF1-F,6#,$F)!\"\"F1F1*(%$_C4GF1F5F1F)F1F1 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Veri fication" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "subs(non_homog_soln, non_homog_eqn);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,.-%%diffG6$-F&6$-F&6$,*-%$expG6#,$%\"xG\"\"$#\" \"\"\"\"%*&%$_C2GF4-F.6#,$F1\"\"#F4F4*&%$_C3GF4-F.6#,$F1!\"\"F4F4*(%$_ C4GF4F8F4F1F4F4F1F1F1F4F(!\"$F-F4F6F5F \+ " 0 "" {MPLTEXT 1 0 24 "simplify(lhs(\")-rhs(\"));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 2 " " }{XPPEDIT 264 1 "diff(y(x),x,x,x)-3*diff(y(x) ,x,x)+4*y(x)=2*x*exp(3*x)-5*exp(3*x)" "/,(-%%diffG6&-%\"yG6#%\"xGF*F*F *\"\"\"*&\"\"$F+-F%6%-F(6#F*F*F*F+!\"\"*&\"\"%F+-F(6#F*F+F+,&*(\"\"#F+ F*F+-%$expG6#*&F-F+F*F+F+F+*&\"\"&F+-F;6#*&F-F+F*F+F+F2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "non_homog_eqn := diff(y(x),x,x,x)-3*diff(y(x) ,x,x)+4*y(x) = 2*x*exp(3*x)-5*exp(3*x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.non_homog_eqnG/,(-%%diffG6$-F(6$-F(6$-%\"yG6#%\"xGF1F1F1\"\" \"F*!\"$F.\"\"%,&*&F1F2-%$expG6#,$F1\"\"$F2\"\"#F7!\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "annihilated_homog_eqn := map(w -> d iff(w,x) - 3*w, map(z -> diff(z,x) - 3*z, non_homog_eqn));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%6annihilated_homog_eqnG/,.-%%diffG6$-F(6$- F(6$-F(6$-F(6$-%\"yG6#%\"xGF5F5F5F5F5\"\"\"F*!\"*F.!#BF,\"#FF0!#CF2\"# O\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "annihilated_homog _soln := dsolve( \" , y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%7ann ihilated_homog_solnG/-%\"yG6#%\"xG,,*&%$_C1G\"\"\"-%$expG6#,$F)\"\"$F- F-*&%$_C2GF--F/6#,$F)\"\"#F-F-*&%$_C3GF--F/6#,$F)!\"\"F-F-*(%$_C4GF-F5 F-F)F-F-*(%$_C5GF-F)F-F.F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "subs(\", non_homog_eqn);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,0- %%diffG6$-F&6$-F&6$,,*&%$_C1G\"\"\"-%$expG6#,$%\"xG\"\"$F/F/*&%$_C2GF/ -F16#,$F4\"\"#F/F/*&%$_C3GF/-F16#,$F4!\"\"F/F/*(%$_C4GF/F8F/F4F/F/*(%$ _C5GF/F4F/F0F/F/F4F4F4F/F(!\"$F-\"\"%F6FGF " 0 "" {MPLTEXT 1 0 12 "simplify(\");" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&%$_C1G\"\"\"-%$expG6#,$%\"xG\"\"$ F'\"\"%*&%$_C5GF'F(F'\"\"**(F0F'F,F'F(F'F.,&*&F,F'F(F'\"\"#F(!\"&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "solve(identity(\",x),\{_C1,_ C5\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%$_C5G#\"\"\"\"\"#/%$_C1G #!#>\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "non_homog_soln := subs(\", annihilated_homog_soln);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/non_homog_solnG/-%\"yG6#%\"xG,,-%$expG6#,$F)\"\"$#!#>\"\")*&%$_C 2G\"\"\"-F,6#,$F)\"\"#F5F5*&%$_C3GF5-F,6#,$F)!\"\"F5F5*(%$_C4GF5F6F5F) F5F5*&F)F5F+F5#F5F9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Verification:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "su bs(non_homog_soln, non_homog_eqn);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# /,0-%%diffG6$-F&6$-F&6$,,-%$expG6#,$%\"xG\"\"$#!#>\"\")*&%$_C2G\"\"\"- F.6#,$F1\"\"#F8F8*&%$_C3GF8-F.6#,$F1!\"\"F8F8*(%$_C4GF8F9F8F1F8F8*&F1F 8F-F8#F8F " 0 "" {MPLTEXT 1 0 24 "simplify(lhs(\")-rhs(\"));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 2 " " } {XPPEDIT 266 1 "diff(y(x),x,x,x)-3*diff(y(x),x,x)+4*y(x)=10*exp(2*x)" "/,(-%%diffG6&-%\"yG6#%\"xGF*F*F*\"\"\"*&\"\"$F+-F%6%-F(6#F*F*F*F+!\" \"*&\"\"%F+-F(6#F*F+F+*&\"#5F+-%$expG6#*&\"\"#F+F*F+F+" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "non_homog_eqn := diff(y(x),x,x,x)-3*diff(y(x) ,x,x)+4*y(x) = 10*exp(2*x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.non _homog_eqnG/,(-%%diffG6$-F(6$-F(6$-%\"yG6#%\"xGF1F1F1\"\"\"F*!\"$F.\" \"%,$-%$expG6#,$F1\"\"#\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "annihilated_homog_eqn := map(z -> diff(z,x) - 2*z, non_homog_eqn );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%6annihilated_homog_eqnG/,,-%% diffG6$-F(6$-F(6$-F(6$-%\"yG6#%\"xGF3F3F3F3\"\"\"F*!\"&F.\"\"%F,\"\"'F 0!\")\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "annihilated_h omog_soln := dsolve( \" , y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% 7annihilated_homog_solnG/-%\"yG6#%\"xG,**&%$_C1G\"\"\"-%$expG6#,$F)\" \"#F-F-*&%$_C2GF--F/6#,$F)!\"\"F-F-*(%$_C3GF-F.F-F)F-F-*(%$_C4GF-F.F-F )F2F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "subs(\", non_homog _eqn);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,.-%%diffG6$-F&6$-F&6$,**&% $_C1G\"\"\"-%$expG6#,$%\"xG\"\"#F/F/*&%$_C2GF/-F16#,$F4!\"\"F/F/*(%$_C 3GF/F0F/F4F/F/*(%$_C4GF/F0F/F4F5F/F4F4F4F/F(!\"$F-\"\"%F6FAFFA,$F 0\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(\");" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&%$_C4G\"\"\"-%$expG6#,$%\"xG\"\"# F'\"\"',$F(\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "solve(id entity(\",x),\{_C4\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#/%$_C4G#\" \"&\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "non_homog_soln \+ := subs(\", annihilated_homog_soln);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%/non_homog_solnG/-%\"yG6#%\"xG,**&%$_C1G\"\"\"-%$expG6#,$F)\"\"#F- F-*&%$_C2GF--F/6#,$F)!\"\"F-F-*(%$_C3GF-F.F-F)F-F-*&F.F-F)F2#\"\"&\"\" $" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 13 "Verification:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "subs(non_homog_sol n, non_homog_eqn);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,.-%%diffG6$-F& 6$-F&6$,**&%$_C1G\"\"\"-%$expG6#,$%\"xG\"\"#F/F/*&%$_C2GF/-F16#,$F4!\" \"F/F/*(%$_C3GF/F0F/F4F/F/*&F0F/F4F5#\"\"&\"\"$F4F4F4F/F(!\"$F-\"\"%F6 FCF#\"#?FA,$F0\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 " simplify(lhs(\")-rhs(\"));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 8 " " }{XPPEDIT 268 1 "diff(y(x),x, x)-4*diff(y(x),x)+13*y(x)=2*cos(3*x)-sin(3*x)" "/,(-%%diffG6%-%\"yG6#% \"xGF*F*\"\"\"*&\"\"%F+-F%6$-F(6#F*F*F+!\"\"*&\"#8F+-F(6#F*F+F+,&*&\" \"#F+-%$cosG6#*&\"\"$F+F*F+F+F+-%$sinG6#*&F>F+F*F+F2" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "Let's see what the solutions of this equation are when th e rhs is 0:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "non_homog_eqn := diff(y(x),x,x)-4*diff(y(x),x)+13*y(x ) = 2*cos(3*x)-sin(3*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.non_hom og_eqnG/,(-%%diffG6$-F(6$-%\"yG6#%\"xGF/F/\"\"\"F*!\"%F,\"#8,&-%$cosG6 #,$F/\"\"$\"\"#-%$sinGF6!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "ode2charEqn(non_homog_eqn, y(x), r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(\"#8\"\"\"%\"rG!\"%*$F'\"\"#F&\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "solve(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&\"\"#\"\"\"%\"IG\"\"$,&F$F%F&!\"$" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Solutions of the associated homogeneous equation are" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 " \+ " }{TEXT 271 7 " " }{TEXT 269 9 " " }{XPPEDIT 270 1 " exp(2*x)*(A*cos(3*x)+B*sin(3*x))" "*&-%$expG6#*&\"\"#\"\"\"%\"xGF(F(,& *&%\"AGF(-%$cosG6#*&\"\"$F(F)F(F(F(*&%\"BGF(-%$sinG6#*&F1F(F)F(F(F(F( " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The a nnihilator of the rhs, " }{MPLTEXT 1 0 19 "2*cos(3*x)-sin(3*x)" } {TEXT -1 10 ", is " }{TEXT 272 3 " " }{XPPEDIT 273 1 "z -> diff (z,x,x)+9*z" ":6#%\"zG7\"6$%)operatorG%&arrowG6\",&-%%diffG6%F$%\"xGF. \"\"\"*&\"\"*F/F$F/F/F)F)" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "annihilator := z -> diff(z,x,x)+9*z;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,annihilatorG:6#%\"zG6\"6$%)operatorG%&arrowGF(, &-%%diffG6%9$%\"xGF1\"\"\"F0\"\"*F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "annihilated_homog_eqn := map(annihilator, non_homog _eqn );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%6annihilated_homog_eqnG/, ,-%%diffG6$-F(6$-F(6$-F(6$-%\"yG6#%\"xGF3F3F3F3\"\"\"F*!\"%F,\"#AF.!#O F0\"$<\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "annihilate d_homog_soln := dsolve( \" , y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%7annihilated_homog_solnG/-%\"yG6#%\"xG,**&%$_C1G\"\"\"-%$cosG6#,$F) \"\"$F-F-*&%$_C2GF--%$sinGF0F-F-*(%$_C3GF-F.F--%$expG6#,$F)\"\"#F-F-*( %$_C4GF-F5F-F9F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "subs( \", non_homog_eqn);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,.-%%diffG6$-F &6$,**&%$_C1G\"\"\"-%$cosG6#,$%\"xG\"\"$F-F-*&%$_C2GF--%$sinGF0F-F-*(% $_C3GF-F.F--%$expG6#,$F2\"\"#F-F-*(%$_C4GF-F6F-F:F-F-F2F2F-F(!\"%F+\"# 8F4FBF8FBF?FB,&F.F>F6!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,**&%$_C1G\"\"\" -%$cosG6#,$%\"xG\"\"$F'\"\"%*&%$_C2GF'-%$sinGF*F'F.*&F&F'F1F'\"#7*&F0F 'F(F'!#7,&F(\"\"#F1!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "solve(identity(\",x),\{_C1,_C2\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #<$/%$_C1G#!\"\"\"#S/%$_C2G#!\"(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "non_homog_soln := subs(\", annihilated_homog_soln);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/non_homog_solnG/-%\"yG6#%\"xG,*-% $cosG6#,$F)\"\"$#!\"\"\"#S-%$sinGF-#!\"(F2*(%$_C3G\"\"\"F+F9-%$expG6#, $F)\"\"#F9F9*(%$_C4GF9F3F9F:F9F9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 13 "Verification:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "subs( non_homog_soln , non_homog_eqn);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#/,.-%%diffG6$-F&6$,*-%$cosG6#,$%\"xG\"\"$#!\"\" \"#S-%$sinGF-#!\"(F3*(%$_C3G\"\"\"F+F:-%$expG6#,$F/\"\"#F:F:*(%$_C4GF: F4F:F;F:F:F/F/F:F(!\"%F+#!#8F3F4#!#\"*F3F8\"#8F@FG,&F+F?F4F2" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "simplify(lhs(\")-rhs(\"));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\" !" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 279 4 " " }{XPPEDIT 280 1 "diff(y(x),x,x)-4*d iff(y(x),x)+13*y(x)=2*exp(2*x)*cos(3*x)" "/,(-%%diffG6%-%\"yG6#%\"xGF* F*\"\"\"*&\"\"%F+-F%6$-F(6#F*F*F+!\"\"*&\"#8F+-F(6#F*F+F+*(\"\"#F+-%$e xpG6#*&F8F+F*F+F+-%$cosG6#*&\"\"$F+F*F+F+" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "non_homog_eqn := diff(y(x), x,x)-4*diff(y(x),x)+13*y(x) = 2*exp(2*x)*cos(3*x);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%.non_homog_eqnG/,(-%%diffG6$-F(6$-%\"yG6#%\"xGF/F/ \"\"\"F*!\"%F,\"#8,$*&-%$cosG6#,$F/\"\"$F0-%$expG6#,$F/\"\"#F0F>" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "In the preceding example we have seen that the solution of the homogeneous equation that is associated to " } {MPLTEXT 1 0 13 "non_homog_eqn" }{TEXT -1 4 " is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 " " }{TEXT 276 7 " " }{TEXT 274 9 " " }{XPPEDIT 275 1 "exp(2*x)*(A*cos( 3*x)+B*sin(3*x))" "*&-%$expG6#*&\"\"#\"\"\"%\"xGF(F(,&*&%\"AGF(-%$cosG 6#*&\"\"$F(F)F(F(F(*&%\"BGF(-%$sinG6#*&F1F(F)F(F(F(F(" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "This time the annihila tor of the rhs, " }{MPLTEXT 1 0 19 "2*exp(2*x)*cos(3*x)" }{TEXT -1 9 ", is " }{TEXT 277 3 " " }{XPPEDIT 278 1 "z -> diff(z,x,x)-4*d iff(z,x)+13*z" ":6#%\"zG7\"6$%)operatorG%&arrowG6\",(-%%diffG6%F$%\"xG F.\"\"\"*&\"\"%F/-F,6$F$F.F/!\"\"*&\"#8F/F$F/F/F)F)" }{TEXT -1 1 ":" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "annihilator := z -> diff(z,x ,x)-4*diff(z,x)+13*z;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,annihilato rG:6#%\"zG6\"6$%)operatorG%&arrowGF(,(-%%diffG6%9$%\"xGF1\"\"\"-F.6$F0 F1!\"%F0\"#8F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "annihil ated_homog_eqn := map(annihilator, non_homog_eqn );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%6annihilated_homog_eqnG/,,-%%diffG6$-F(6$-F(6$-F (6$-%\"yG6#%\"xGF3F3F3F3\"\"\"F*!\")F,\"#UF.!$/\"F0\"$p\"\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "annihilated_homog_soln := ds olve( \" , y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%7annihilated_ho mog_solnG/-%\"yG6#%\"xG,**(%$_C1G\"\"\"-%$cosG6#,$F)\"\"$F--%$expG6#,$ F)\"\"#F-F-*(%$_C2GF--%$sinGF0F-F3F-F-**%$_C3GF-F.F-F3F-F)F-F-**%$_C4G F-F:F-F3F-F)F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "subs(\" , non_homog_eqn);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,.-%%diffG6$-F&6 $,**(%$_C1G\"\"\"-%$cosG6#,$%\"xG\"\"$F--%$expG6#,$F2\"\"#F-F-*(%$_C2G F--%$sinGF0F-F4F-F-**%$_C3GF-F.F-F4F-F2F-F-**%$_C4GF-F;F-F4F-F2F-F-F2F 2F-F(!\"%F+\"#8F9FBF=FBF?FB,$*&F.F-F4F-F8" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 12 "simplify(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/, &*(%$_C4G\"\"\"-%$cosG6#,$%\"xG\"\"$F'-%$expG6#,$F,\"\"#F'\"\"'*(%$_C3 GF'-%$sinGF*F'F.F'!\"',$*&F(F'F.F'F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "solve(identity(\",x),\{_C3,_C4\});" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#<$/%$_C3G\"\"!/%$_C4G#\"\"\"\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "non_homog_soln := subs(\", annihilated_homo g_soln);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/non_homog_solnG/-%\"yG6 #%\"xG,(*(%$_C1G\"\"\"-%$cosG6#,$F)\"\"$F--%$expG6#,$F)\"\"#F-F-*(%$_C 2GF--%$sinGF0F-F3F-F-*(F:F-F3F-F)F-#F-F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Verification:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "subs( non_homog_soln , non_homog_eqn);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,,-%%diffG6$-F&6$,(*(%$_C1G\"\"\"-%$cosG6#, $%\"xG\"\"$F--%$expG6#,$F2\"\"#F-F-*(%$_C2GF--%$sinGF0F-F4F-F-*(F;F-F4 F-F2F-#F-F3F2F2F-F(!\"%F+\"#8F9F@F=#F@F3,$*&F.F-F4F-F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "simplify(lhs(\")-rhs(\"));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 23 "Variation of Parameters" }}{PARA 0 "" 0 " " {TEXT -1 5 " If " }{TEXT 281 2 " " }{XPPEDIT 282 1 "y[1](x)" "-&% \"yG6#\"\"\"6#%\"xG" }{TEXT -1 7 " and " }{TEXT 283 2 " " } {XPPEDIT 284 1 "y[2](x)" "-&%\"yG6#\"\"#6#%\"xG" }{TEXT 285 2 " " } {TEXT -1 53 "are independent solutions of the homogeneous equation" } {TEXT 286 3 " " }{XPPEDIT 287 1 "Diff(y(x),x,x)+a[1]*Diff(y(x),x)+a[ 0*y(x) = 0" ",(-%%DiffG6%-%\"yG6#%\"xGF)F)\"\"\"*&&%\"aG6#F*F*-F$6$-F' 6#F)F)F*F*&F-6#/*&\"\"!F*-F'6#F)F*F7F*" }{TEXT -1 19 " , then \n \+ " }{TEXT 288 12 " " }{XPPEDIT 289 1 "y(x) = alpha*y[1]( x)+beta*y[2](x)-y[1](x)*Int(y[2](x)*f(x)/W(x),x)+y[2](x)*Int(y[1](x)*f (x)/W(x),x)" "/-%\"yG6#%\"xG,**&%&alphaG\"\"\"-&F$6#F*6#F&F*F**&%%beta GF*-&F$6#\"\"#6#F&F*F**&-&F$6#F*6#F&F*-%$IntG6$*(-&F$6#F46#F&F*-%\"fG6 #F&F*-%\"WG6#F&!\"\"F&F*FI*&-&F$6#F46#F&F*-F<6$*(-&F$6#F*6#F&F*-FD6#F& F*-FG6#F&FIF&F*F*" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 39 "is the general solution of the equation" }{TEXT 290 2 " " }{XPPEDIT 291 1 "Diff(y(x),x,x)+a[1]*Diff(y(x),x)+a[ 0]*y(x) = f(x)" "/,(-%%DiffG6%-%\"yG6#%\"xGF*F*\"\"\"*&&%\"aG6#F+F+-F% 6$-F(6#F*F*F+F+*&&F.6#\"\"!F+-F(6#F*F+F+-%\"fG6#F*" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1191 "variationOfParameters := proc()\n local x,y, y1,y2, W;\n if \+ nargs <> 4 then ERROR(`variationOfParameters expects four arguments.`) ;\n elif not type(args[1],\{name,procedure\}) or not type(args[2],\{ name,procedure\}) then\n ERROR(`variationOfParameters expects e ach of its first two arguments to be a\nname or procedure.`);\n elif not type(args[3], equation) then\n ERROR(`variationOfParameters \+ expects its third argument to be an equation`);\n elif not type(args [4],function) then\n ERROR(`variationOfParameters expects its fou rth argument to be an function`);\n elif not testeq(args[4] = op(0, args[4])(op(1,args[4]))) then\n ERROR(`variationOfParameters expe cts its fourth argument to be a function of one variable.`);\n fi;\n x := op(1,args[4]);\n y := op(0,args[4]);\n y1 := args[1];\n \+ y2 := args[2];\nif not testeq(subs(y(x)=y1(x),lhs(args[3]))=0) or not \+ testeq(subs(y(x)=y2(x),lhs(args[3]))=0) then\n ERROR(`variationOfPar ameters expects each of its first two arguments to be a\nsolution of i ts third argument.`);\nfi;\nW := y1(x)*diff(y2(x),x) - y2(x)*diff(y1(x ),x);\nRETURN(y(x) = -y1(x)*Int(y2(x)*rhs(args[3])/W,x) + y2(x)*Int(y1 (x)*rhs(args[3])/W,x));\nend; \n " }}{PARA 12 "" 1 "" {XPPMATH 20 "6 #>%6variationOfParametersG:6\"6'%\"xG%\"yG%#y1G%#y2G%\"WGF&F&C*@,09#\" \"%-%&ERRORG6#%NvariationOfParameters~expects~four~arguments.G43-%%typ eG6$&9\"6#\"\"\"<$%%nameG%*procedureG-F96$&F<6#\"\"#F?-F36#%cpvariatio nOfParameters~expects~each~of~its~first~two~arguments~to~be~a|+name~or ~procedure.G4-F96$&F<6#\"\"$%)equationG-F36#%]ovariationOfParameters~e xpects~its~third~argument~to~be~an~equationG4-F96$&F<6#F1%)functionG-F 36#%^ovariationOfParameters~expects~its~fourth~argument~to~be~an~funct ionG4-%'testeqG6#/FW--%#opG6$\"\"!FW6#-F^o6$F>FW-F36#%^pvariationOfPar ameters~expects~its~fourth~argument~to~be~a~function~of~one~variable.G >8$Fbo>8%F]o>8&F;>8'FD@$43-Fin6#/-%%subsG6$/-Fjo6#Fho-F\\pFjp-%$lhsG6# FMF`o-Fin6#/-Ffp6$/Fip-F^pFjpF\\qF`o-F36#%`qvariationOfParameters~expe cts~each~of~its~first~two~arguments~to~be~a|+solution~of~its~third~arg ument.G>8(,&*&F[qF>-%%diffG6$FeqFhoF>F>*&FeqF>-F^r6$F[qFhoF>!\"\"-%'RE TURNG6#/Fip,&*&F[qF>-%$IntG6$*(FeqF>-%$rhsGF^qF>FjqFcrFhoF>Fcr*&FeqF>- F[s6$*(F[qF>F^sF>FjqFcrFhoF>F>F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "ode := diff(y(x),x$2) + y(x) = tan(x);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$odeG/,&-%%diffG6$-F(6$-%\"yG6#%\"xGF/F/\"\"\" F,F0-%$tanGF." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "variationO fParameters(cos,sin,ode,y(x)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% \"yG6#%\"xG,&*&-%$cosGF&\"\"\"-%$IntG6$*(-%$sinGF&F,-%$tanGF&F,,&*$F* \"\"#F,*$F1F7F,!\"\"F'F,F9*&F1F,-F.6$*(F*F,F3F,F5F9F'F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&-%$cosGF&\"\"\",&-%$sinGF&!\"\"-%#lnG 6#,&-%$secGF&F,-%$tanGF&F,F,F,F0*&F.F,F*F,F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "subs(\",ode);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/,(-%%diffG6$-F&6$,&*&-%$cosG6#%\"xG\"\"\",&-%$sinGF.!\"\"-%#lnG6#,&- %$secGF.F0-%$tanGF.F0F0F0F4*&F2F0F,F0F4F/F/F0F+F4F=F4F;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "testeq(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 362 17 "Notice here that " }{TEXT 363 18 "names of function s" }{TEXT 364 47 " have been passed as the first two arguments of" } {TEXT -1 2 " " }{MPLTEXT 1 0 21 "variationOfParameters" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT 365 44 "The functions themselves can also b e passed:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "variationOfParameters(fred -> cos(fred), ethel -> sin (ethel), ode, y(x)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"x G,&*&-%$cosGF&\"\"\"-%$IntG6$*(-%$sinGF&F,-%$tanGF&F,,&*$F*\"\"#F,*$F1 F7F,!\"\"F'F,F9*&F1F,-F.6$*(F*F,F3F,F5F9F'F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 366 84 "But you cannot pass the \+ expressions that define the functions - programmer's choice:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "va riationOfParameters(cos(x), sin(x), ode, y(x));" }}{PARA 8 "" 1 "" {TEXT -1 122 "Error, (in variationOfParameters) variationOfParameters \+ expects each of its first two arguments to be a\nname or procedure." } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 32 "Copyright and Author Information " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 44 "2.5 epR4.mws A Maple Release 4 worksheet." }}{PARA 261 "" 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" {TEXT -1 24 "Author: Brian E. Blank " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT -1 22 "Date: \+ 13 October 2000" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "This document may not be distributed by any medium," }} {PARA 0 "" 0 "" {TEXT -1 55 "including print, disk, and electronic tra nsfer, without" }}{PARA 0 "" 0 "" {TEXT -1 39 "prior written permissio n of the author." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT -1 49 "For more information, please contact the author:" }} {PARA 265 "" 0 "" {TEXT -1 4 " " }}{PARA 265 "" 0 "" {TEXT -1 32 " \+ Department of Mathematics, " }}{PARA 0 "" 0 "" {TEXT -1 39 " W ashington University in St. Louis" }}{PARA 0 "" 0 "" {TEXT -1 26 " \+ St. Louis, MO 63130" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 " " 0 "" {TEXT -1 33 " Telephone: (314) 935-6763" }}{PARA 266 "" 0 "" {TEXT -1 44 " e-mail: brian@math.wustl.edu" }} {PARA 267 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Copyrig ht: \251 2000 Brian E. Blank, All Rights Reserved." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}}{MARK "7" 0 }{VIEWOPTS 1 1 0 3 4 1802 }