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\"\"F,*&F*F,-%#lnG6#,&F*F,-F&6#,&*$F*F+F,*$F.F+F/F,F,F/*(F*F,F2F,F.F,F ,%\"xGF,%\"CG" }{TEXT -1 120 "\nis an implicit solution of the differe ntial equation\n \+ " }{TEXT 258 2 "y'" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "-y/s qrt(a^2-y^2)" "6#,$*&%\"yG\"\"\"-%%sqrtG6#,&*$%\"aG\"\"#F&*$F%F-!\"\"F /F/" }{TEXT -1 82 "\nwhich describes the trajectory of an object that \+ is pulled with a rope of length " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 32 " by someone who walks along the " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 20 "-axis to the right.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "alias(y=y(x)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "implicit Solution := sqrt(a^2-y^2) -\n a*ln(a+sqrt(a^2-y^2)) + a*ln(y) + x = \+ C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%1implicitSolutionG/,**$,&*$)% \"aG\"\"#\"\"\"F-*$)%\"yGF,F-!\"\"#F-F,F-*&F+F--%#lnG6#,&F+F-F'F-F-F1* &F+F--F56#F0F-F-%\"xGF-%\"CG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "diff(implicitSolution, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/, **(,&*$)%\"aG\"\"#\"\"\"F+*$)%\"yGF*F+!\"\"#F/F*F.F+-%%diffG6$F.%\"xGF +F/*,F)F+F&F0F.F+F1F+,&F)F+*$F&#F+F*F+F/F+*(F)F+F1F+F.F/F+F+F+\"\"!" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Diff(y,x) = solve(%, diff( y,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$%\"yG%\"xG,$*(,&% \"aG\"\"\"*$,&*$)F,\"\"#F-F-*$)F'F2F-!\"\"#F-F2F-F-F'F-,(F3F5F0F-*&F,F -F/F6F-F5F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "subs(y^2=a^2 -z^2,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$%\"yG%\"xG,$*(, &%\"aG\"\"\"*$*$)%\"zG\"\"#F-#F-F2F-F-F'F-,&F/F-*&F,F-F/F3F-!\"\"F6" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "simplify(%,symbolic);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$%\"yG%\"xG,$*&F'\"\"\"%\"zG !\"\"F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "subs(z=sqrt(a^2- y^2), %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$%\"yG%\"xG,$*& F'\"\"\",&*$)%\"aG\"\"#F+F+*$)F'F0F+!\"\"#F3F0F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 272 2 "2. " }{TEXT -1 150 " Solve the following ODEs with Maple. Try several met hods, try to find the solutions in their simplest form, and check if M aple finds all solutions.\n\n" }{TEXT 273 3 "(a)" }{TEXT -1 1 " " } {XPPEDIT 18 0 "3y^2" "6#*&\"\"$\"\"\"*$%\"yG\"\"#F%" }{TEXT 259 2 "y' " }{TEXT -1 3 " + " }{XPPEDIT 18 0 "16*x=12*x*y^3" "6#/*&\"#;\"\"\"%\" xGF&*(\"#7F&F'F&%\"yG\"\"$" }{TEXT -1 3 ".\n\n" }{TEXT 274 3 "(b)" } {TEXT -1 1 " " }{TEXT 275 2 "y'" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "2*y /x-(y/x)^2" "6#,&*(\"\"#\"\"\"%\"yGF&%\"xG!\"\"F&*$*&F'F&F(F)F%F)" } {TEXT -1 3 ".\n\n" }{TEXT 276 3 "(c)" }{TEXT -1 1 " " }{TEXT 260 3 "xy '" }{TEXT -1 3 " - " }{XPPEDIT 18 0 "y=x*tan(y/x)" "6#/%\"yG*&%\"xG\" \"\"-%$tanG6#*&F$F'F&!\"\"F'" }{TEXT -1 2 ".\n" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 302 3 "(a)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "rest art;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "alias(y=y(x)):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "ODE := 3*y^2*diff(y,x) + 16* x = 12*x*y^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ODEG/,&*(\"\"$\"\" \")%\"yG\"\"#F)-%%diffG6$F+%\"xGF)F)*&\"#;F)F0F)F),$*(\"#7F)F0F))F+F(F )F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "dsolve(ODE, y);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%/%\"yG,$*&#\"\"\"\"\"$F(*$),&\"#OF(*( \"#FF(-%$expG6#,$*&\"\"'F()%\"xG\"\"#F(F(F(%$_C1GF(F(F'F(F(F(/F$,&*&#F (F5F(F*F(!\"\"*(^##F(F5F(F)#F(F8F+F(F(/F$,&*&#F(F5F(F*F(F>*(^##F>F5F(F )FBF+F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "sol := %[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG/%\"yG,$*&#\"\"\"\"\"$F**$),& \"#OF**(\"#FF*-%$expG6#,$*&\"\"'F*)%\"xG\"\"#F*F*F*%$_C1GF*F*F)F*F*F* " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "testeq(eval(ODE, sol)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 14 "Alternatively:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "ode test(sol, ODE);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 303 3 "(b)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "alias(y= y(x)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "ODE := diff(y,x) \+ = 2*y/x - (y/x)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ODEG/-%%diffG 6$%\"yG%\"xG,&*(\"\"#\"\"\"F)F.F*!\"\"F.*&F)F-F*!\"#F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "sol := dsolve(ODE, y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG/%\"yG*&%\"xG\"\"#,&F(\"\"\"%$_C1GF+!\" \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "testeq(eval(ODE, sol) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "odetest(sol, ODE);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 301 3 "(c)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "alias(y=y(x)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "ODE := x *diff(y,x) - y = x*tan(y/x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ODE G/,&*&%\"xG\"\"\"-%%diffG6$%\"yGF(F)F)F-!\"\"*&F(F)-%$tanG6#*&F-F)F(F. F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "sol := dsolve(ODE, y) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG/%\"yG*&-%'arcsinG6#*&%\"x G\"\"\"%$_C1GF-F-F,F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "te steq(eval(ODE, sol));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%FAILG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eval(ODE, sol);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&%\"xG\"\"\",&*(%$_C1GF',&F'F'*&)F&\"\"#F') F*F.F'!\"\"#F0F.F&F'F'-%'arcsinG6#*&F&F'F*F'F'F'F'*&F2F'F&F'F0*(F&F.F* F'F+F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%\"xG\"\"#%$_C1G\"\"\",&F(F(*&)F%F& F()F'F&F(!\"\"#F-F&F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "od etest(sol, ODE);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 277 3 "3. " }{TEXT -1 100 "Some ODE's of degree larger than one \+ can be solved with Maple. Consider the following two examples.\n" } {TEXT 278 3 "(a)" }{TEXT -1 7 " Solve " }{XPPEDIT 18 0 "`y' `^2" "6#*$ %$y'~G\"\"#" }{TEXT -1 3 " - " }{XPPEDIT 18 0 "y^2=0" "6#/*$%\"yG\"\"# \"\"!" }{TEXT -1 3 ".\n\n" }{TEXT 279 3 "(b)" }{TEXT -1 7 " Solve " } {XPPEDIT 18 0 "`y' `^2" "6#*$%$y'~G\"\"#" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "+x*y=y^2+x" "6#/*&%\"xG\"\"\"%\"yGF&,&*$F'\"\"#F&F%F&" }{TEXT 261 4 "y' \n" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 305 3 "(a)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 14 "alias(y=y(x)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "ODE := diff(y,x)^2 - y^2 = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ODEG/,&*$)-%%diffG6$%\"yG%\"xG\"\"#\"\"\"F/*$)F,F.F/ !\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "sols := dsolv e(ODE, y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solsG6$/%\"yG*&%$_C1G \"\"\"-%$expG6#%\"xGF*/F'*&F)F*-F,6#,$F.!\"\"F*" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 27 "testeq(eval(ODE, sols[1]));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "testeq(eval(ODE, sols[2]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%true G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 " " 0 "" {TEXT 304 3 "(b)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "re start;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "alias(y=y(x)):" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "ODE := diff(y,x)^2 + x*y = y^2 +x*diff(y,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ODEG/,&*$)-%% diffG6$%\"yG%\"xG\"\"#\"\"\"F/*&F-F/F,F/F/,&*$)F,F.F/F/*&F-F/F)F/F/" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "sols := dsolve(ODE, y);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solsG6$/%\"yG,(%\"xG\"\"\"F*!\"\"* &-%$expG6#,$F)F+F*%$_C1GF*F*/F'*&F1F*-F.6#F)F*" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 27 "testeq(eval(ODE, sols[1]));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "testeq(eval(ODE, sols[2]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%true G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 280 2 "4." }{TEXT -1 60 " Solve the following ODEs with \+ Maple and check the answers.\n" }{TEXT 281 3 "(a)" }{TEXT -1 1 " " } {XPPEDIT 18 0 "x^4" "6#*$%\"xG\"\"%" }{TEXT 262 3 "y''" }{TEXT -1 3 " \+ - " }{XPPEDIT 18 0 "(2x^2-1)*x" "6#*&,&*&\"\"#\"\"\"*$%\"xGF&F'F'F'!\" \"F'F)F'" }{TEXT 263 2 "y'" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "y=0" "6# /%\"yG\"\"!" }{TEXT -1 3 ".\n\n" }{TEXT 282 3 "(b)" }{TEXT -1 1 " " } {XPPEDIT 18 0 "x^2" "6#*$%\"xG\"\"#" }{TEXT -1 6 "y'' + " }{XPPEDIT 18 0 "3x" "6#*&\"\"$\"\"\"%\"xGF%" }{TEXT 264 2 "y'" }{TEXT -1 3 " + \+ " }{XPPEDIT 18 0 "(x^2-35)*y=x" "6#/*&,&*$%\"xG\"\"#\"\"\"\"#N!\"\"F)% \"yGF)F'" }{TEXT -1 3 ".\n\n" }{TEXT 283 3 "(c)" }{TEXT -1 1 " " } {TEXT 265 2 "y'" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "x*y^2=1" "6#/*&%\"x G\"\"\"*$%\"yG\"\"#F&F&" }{TEXT -1 2 ".\n" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 306 3 "(a)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "alias(y=y(x)):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "ODE := x^4*diff(y,x$2) - (2* x^2-1)*x*diff(y,x) + y = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ODEG /,(*&)%\"xG\"\"%\"\"\"-%%diffG6$%\"yG-%\"$G6$F)\"\"#F+F+*(,&*&F3F+)F)F 3F+F+F+!\"\"F+F)F+-F-6$F/F)F+F8F/F+\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "sol := dsolve(ODE, y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG/%\"yG,&*(%$_C1G\"\"\"%\"xG!\"\",(F*F,*&\"\"#F*)F+F/F*F** $)F+\"\"%F*F*F*F*,$-%$IntG6$*(F+F3F-!\"#-%$expG6#,$*&F*F**&F/F*F0F*F,F *F*F+*(%$_C2GF*F+F,F-F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "testeq(simplify(subs(sol, ODE)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 307 3 "(b)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "alias(y= y(x)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "ODE := x^2*diff(y ,x$2) +3*x*diff(y,x) + (x^2-35)*y = x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ODEG/,(*&)%\"xG\"\"#\"\"\"-%%diffG6$%\"yG-%\"$G6$F)F*F+F+*(\" \"$F+F)F+-F-6$F/F)F+F+*&,&*$F(F+F+\"#N!\"\"F+F/F+F+F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "sol := dsolve(ODE, y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG/%\"yG,(*(%\"xG!\"\"-%(BesselJG6$\"\"'F) \"\"\"%$_C2GF/F/*(F)F*-%(BesselYGF-F/%$_C1GF/F/*&,*\"&SI#F/*&\"%_6F/)F )\"\"#F/F/*&\"#OF/)F)\"\"%F/F/*$)F)F.F/F/F/F)!\"(F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "testeq(simplify(subs(sol, ODE)));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 308 3 "(c)" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 14 "alias(y=y(x)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "ODE := diff(y,x) + x*y^2 = 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ODEG/,&-%%diffG6$%\"yG%\"xG\"\"\"*&F+F,)F*\"\"#F,F,F ," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "sol := dsolve(ODE, y); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG/%\"yG*(,&-%(BesselKG6$#\" \"\"\"\"$,$*(\"\"#F-F.!\"\"%\"xG#F.F1F-F2*&%$_C1GF--%(BesselIG6$#F2F.F /F-F-F-F3#F2F1,&*&F6F--F86$#F1F.F/F-F--F*F?F-F2" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 25 "simplify(subs(sol, ODE));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"\"F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 284 2 "5." }{TEXT -1 73 " Conside r the initial value problem\n " } {XPPEDIT 18 0 "`y'' `-y=0" "6#/,&%%y''~G\"\"\"%\"yG!\"\"\"\"!" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 4 ", " }{TEXT 266 2 "y'" }{TEXT -1 11 " (0) = 0.\n\n" }{TEXT 285 3 "(a)" }{TEXT -1 58 " Find the solutions via the method of Laplace tr ansforms.\n" }{TEXT 286 3 "(b)" }{TEXT -1 31 " Redo (a) after you have given " }{TEXT 0 17 "infolevel[dsolve]" }{TEXT -1 14 " the value 3.\n " }{TEXT 287 3 "(c)" }{TEXT -1 26 " Repeat (a), but now with " }{TEXT 0 10 "printlevel" }{TEXT -1 15 " equal to 33.\n" }}{SECT 1 {PARA 0 " " 0 "" {TEXT 295 3 "(a)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "re start;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "ODE := diff(y(x), x$2) - y(x) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ODEG/,&-%%diff G6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"F*!\"\"\"\"!" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 30 "initvals := y(0)=1, D(y)(0)=0;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%)initvalsG6$/-%\"yG6#\"\"!\"\"\"/--%\"DG6#F(F) F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "dsolve(\{ODE, initval s\}, y(x), method=laplace);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG 6#%\"xG-%%coshGF&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 0 "" 0 "" {TEXT 296 3 "(b)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "ODE := diff(y(x), x$2) - y(x) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%$ODEG/,&-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"F*!\"\"\"\"!" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "initvals := y(0)=1, D(y)(0 )=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)initvalsG6$/-%\"yG6#\"\"!\" \"\"/--%\"DG6#F(F)F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "inf olevel [dsolve] := 3:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "ds olve(\{ODE, initvals\}, y(x), method=laplace);" }}{PARA 6 "" 1 "" {TEXT -1 125 "dsolve/inttrans/solveit: Transform of eqns is \{_s1* (_s1*`laplace/internal`(y(x),x,_s1)-1)-`laplace/internal`(y(x),x,_s1) \}" }}{PARA 6 "" 1 "" {TEXT -1 99 "dsolve/inttrans/solveit: Algebrai c Solution is \{`laplace/internal`(y(x),x,_s1) = _s1/(_s1^2-1)\}" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%%coshGF&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 297 3 "(c)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "ODE := diff(y(x), x$2) - y(x ) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ODEG/,&-%%diffG6$-%\"yG6# %\"xG-%\"$G6$F-\"\"#\"\"\"F*!\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "initvals := y(0)=1, D(y)(0)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)initvalsG6$/-%\"yG6#\"\"!\"\"\"/--%\"DG6#F(F)F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "printlevel := 33:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "dsolve(\{ODE, initvals\}, y( x), method=laplace);" }}{PARA 9 "" 1 "" {TEXT -1 22 "\{--> enter D, ar gs = y" }}{PARA 9 "" 1 "" {TEXT -1 39 "\{--> enter D/_procname, args = D, Index" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG%\"DG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%&IndexG6\"" }}{PARA 9 "" 1 "" {TEXT -1 36 "<-- exit D/_procname (now in D) = D\}" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%$OprG%\"DG" }}{PARA 9 "" 1 "" {TEXT -1 49 "value remembered (in D): \+ is(y, constant) -> false" }}{PARA 9 "" 1 "" {TEXT -1 35 "\{--> enter t ype/D/partial, args = y" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" } }{PARA 9 "" 1 "" {TEXT -1 43 "<-- exit type/D/partial (now in D) = fal se\}" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG%\"yG" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$idxG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG% $D/yG" }}{PARA 9 "" 1 "" {TEXT -1 36 "\{--> enter D/_diff, args = y, x , [x]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%xvarG%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"dG%'diff/yG" }}{PARA 9 "" 1 "" {TEXT -1 35 "<-- exit D/_diff (now in D) = FAIL\}" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG%%FAILG" }} {PARA 9 "" 1 "" {TEXT -1 37 "<-- exit D (now at top level) = D(y)\}" } }{PARA 9 "" 1 "" {TEXT -1 38 "\{--> enter evalapply, args = D(y), [0] " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG%\"DG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG%,evalapply/DG" }}{PARA 9 "" 1 "" {TEXT -1 40 "\{ --> enter evalapply/D, args = D(y), [0]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'AssertG6#%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'Assert G6#/%\"DGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#--%\"DG6#%\"yG6#\"\"!" }}{PARA 9 "" 1 "" {TEXT -1 50 "<-- exit evalapply/D (now in evalapply) = D(y)(0)\}" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#--%\"DG6#%\"yG6#\"\"! " }}{PARA 9 "" 1 "" {TEXT -1 48 "<-- exit evalapply (now at top level) = D(y)(0)\}" }}{PARA 9 "" 1 "" {TEXT -1 104 "\{--> enter dsolve, args = \{diff(diff(y(x),x),x)-y(x) = 0, y(0) = 1, D(y)(0) = 0\}, y(x), met hod = laplace" }}{PARA 9 "" 1 "" {TEXT -1 91 "\{--> enter type/ODEtool s/ODE, args = \{diff(diff(y(x),x),x)-y(x) = 0, y(0) = 1, D(y)(0) = 0\} " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}{PARA 9 "" 1 "" {TEXT -1 51 "<-- exit type/ODEtools/ODE (now in dsolve) = false\}" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ARGSG7$-%\"yG6#%\"xG/%'methodG%(laplaceG " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'METHODG7$/%'methodG%(laplaceG%0 dsolve/INTTRANSG" }}{PARA 9 "" 1 "" {TEXT -1 113 "\{--> enter dsolve/I NTTRANS, args = \{diff(diff(y(x),x),x)-y(x) = 0, y(0) = 1, D(y)(0) = 0 \}, y(x), method = laplace" }}{PARA 9 "" 1 "" {TEXT -1 134 "\{--> ente r dsolve/process_input, args = \{diff(diff(y(x),x),x)-y(x) = 0, y(0) = 1, D(y)(0) = 0\}, y(x), deqns, svars, invar, inits, eqns" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqnsG<%/,&-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F. \"\"#\"\"\"F+!\"\"\"\"!/-F,6#F5F3/--%\"DG6#F,F8F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&svarsG<#-%\"yG6#%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"iG-%\"yG6#%\"xG" }}{PARA 9 "" 1 "" {TEXT -1 31 "\{--> enter \+ unknown, args = y(x)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"xG" }} {PARA 9 "" 1 "" {TEXT -1 51 "<-- exit unknown (now in dsolve/process_i nput) = x\}" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&invarG<#%\"xG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&invarG%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&deqnsG<\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&init sG<\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG/,&-%%diffG6$-%\"yG6#% \"xG-%\"$G6$F-\"\"#\"\"\"F*!\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&deqnsG<#,&-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"F*!\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG/-%\"yG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$fcnG%\"yG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"iG%\"yG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&initsG<#/-%\"y G6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG/--%\"DG6#%\"y G6#\"\"!F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$fcnG-%\"DG6#%\"yG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"iG%\"yG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&initsG<$/-%\"yG6#\"\"!\"\"\"/--%\"DG6#F(F)F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%indsG<\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"iG-%\"yG6#%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%&deqnsG<#,&-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"F*!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&svarsG<#-%\"yG6#%\"xG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%&invarG%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%&initsG<$/-%\"yG6#\"\"!\"\"\"/--%\"DG6#F(F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqnsG<%/,&-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F.\"\"#\"\" \"F+!\"\"\"\"!/-F,6#F5F3/--%\"DG6#F,F8F5" }}{PARA 9 "" 1 "" {TEXT -1 112 "<-- exit dsolve/process_input (now in dsolve/INTTRANS) = \{diff(d iff(y(x),x),x)-y(x) = 0, y(0) = 1, D(y)(0) = 0\}\}" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#<%/,&-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F,\"\"#\"\"\"F)! \"\"\"\"!/-F*6#F3F1/--%\"DG6#F*F6F3" }}{PARA 9 "" 1 "" {TEXT -1 177 " \{--> enter dsolve/process_options, args = \{diff(diff(y(x),x),x)-y(x) = 0, y(0) = 1, D(y)(0) = 0\}, x, dtype, implicit, dmethod, dchoice, d form, dpoint, Rest, pt, method = laplace" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&dtypeG%&exactG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)implicit G%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(dmethodG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(dchoiceG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&dformG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'dpointG\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%RestG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(inttranG<(%'hankelG%(fourierG%(hilbertG%(laplaceG%+f ouriercosG%+fouriersinG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%restG/%' methodG%(laplaceG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"iG/%'methodG% (laplaceG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ileftG%'methodG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'irightG%(laplaceG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(dmethodG%(laplaceG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG/,&-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"F*!\"\"\"\" !" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG/-%\"yG6#\"\"!\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ptG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG/--%\"DG6#%\"yG6#\"\"!F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ptG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&dtype G%&exactG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)implicitG%%trueG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%(dmethodG%(laplaceG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(dchoiceG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%&dformG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'dpointG\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%RestG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ptG\"\"!" }}{PARA 9 "" 1 "" {TEXT -1 61 "<-- exit ds olve/process_options (now in dsolve/INTTRANS) = 0\}" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"\"!" }}{PARA 9 "" 1 "" {TEXT -1 107 "\{--> enter ds olve/inttrans, args = laplace, \{diff(diff(y(x),x),x)-y(x)\}, \{y(x)\} , x, \{y(0) = 1, D(y)(0) = 0\}" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&i nitsG<$-%\"yG6#\"\"!--%\"DG6#F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#ptG<#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ptG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&deqnsG<#,&-%%diffG6$-%\"yG6#%\"xG-%\"$G6$ F-\"\"#\"\"\"F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&initsG<$/-% \"yG6#\"\"!\"\"\"/--%\"DG6#F(F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"iG<\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"iG<\"" }}{PARA 9 "" 1 "" {TEXT -1 115 "\{--> enter dsolve/inttrans/solveit, args = laplace, \+ \{diff(diff(y(x),x),x)-y(x)\}, \{y(x)\}, x, \{y(0) = 1, D(y)(0) = 0\} " }}{PARA 9 "" 1 "" {TEXT -1 48 "\{--> enter tools/genglobal, args = _ s, [], reset" }}{PARA 9 "" 1 "" {TEXT -1 61 "<-- exit tools/genglobal \+ (now in dsolve/inttrans/solveit) = \}" }}{PARA 9 "" 1 "" {TEXT -1 93 " \{--> enter `tools/genglobal`[1], args = _s, [laplace, \{diff(diff(y(x ),x),x)-y(x)\}, \{y(x)\}, x]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+Exc eptionsG<$%\"xG%(laplaceG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*candid ateG%$_s1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%$_s1G" }}{PARA 9 "" 1 " " {TEXT -1 69 "<-- exit `tools/genglobal`[1] (now in dsolve/inttrans/s olveit) = _s1\}" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG%$_s1G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%*transnameG%(laplaceG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%'tindexG6\"" }}{PARA 9 "" 1 "" {TEXT -1 54 "\{ --> enter type/De, args = \{diff(diff(y(x),x),x)-y(x)\}" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%&falseG" }}{PARA 9 "" 1 "" {TEXT -1 58 "<-- exi t type/De (now in dsolve/inttrans/solveit) = false\}" }}{PARA 9 "" 1 " " {TEXT -1 52 "\{--> enter type/De, args = diff(diff(y(x),x),x)-y(x)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}{PARA 9 "" 1 "" {TEXT -1 58 "<-- exit type/De (now in dsolve/inttrans/solveit) = false\}" }} {PARA 9 "" 1 "" {TEXT -1 47 "\{--> enter type/De, args = diff(diff(y(x ),x),x)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}{PARA 9 "" 1 " " {TEXT -1 58 "<-- exit type/De (now in dsolve/inttrans/solveit) = fal se\}" }}{PARA 9 "" 1 "" {TEXT -1 39 "\{--> enter type/De, args = diff( y(x),x)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}{PARA 9 "" 1 " " {TEXT -1 58 "<-- exit type/De (now in dsolve/inttrans/solveit) = fal se\}" }}{PARA 9 "" 1 "" {TEXT -1 31 "\{--> enter type/De, args = y(x) " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}{PARA 9 "" 1 "" {TEXT -1 58 "<-- exit type/De (now in dsolve/inttrans/solveit) = false\}" }} {PARA 9 "" 1 "" {TEXT -1 28 "\{--> enter type/De, args = x" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}{PARA 9 "" 1 "" {TEXT -1 58 "<-- exit type/De (now in dsolve/inttrans/solveit) = false\}" }}{PARA 9 " " 1 "" {TEXT -1 32 "\{--> enter type/De, args = -y(x)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}{PARA 9 "" 1 "" {TEXT -1 58 "<-- exit \+ type/De (now in dsolve/inttrans/solveit) = false\}" }}{PARA 9 "" 1 "" {TEXT -1 29 "\{--> enter type/De, args = -1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}{PARA 9 "" 1 "" {TEXT -1 58 "<-- exit type/ De (now in dsolve/inttrans/solveit) = false\}" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t1G<$-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F,\"\"#-F'6$F)F, " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t1G<%-%%diffG6$-%\"yG6#%\"xG-% \"$G6$F,\"\"#F)-F'6$F)F," }}{PARA 9 "" 1 "" {TEXT -1 100 "\{--> enter \+ type/linear, args = diff(diff(y(x),x),x)-y(x), \{y(x), diff(y(x),x), d iff(diff(y(x),x),x)\}" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }} {PARA 9 "" 1 "" {TEXT -1 61 "<-- exit type/linear (now in dsolve/inttr ans/solveit) = true\}" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t2G<\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&neqnsG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t1G,&-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F,\"\"#\"\"\"F)! \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$indG<%-%%diffG6$-%\"yG6#%\" xG-%\"$G6$F,\"\"#F)-F'6$F)F," }}{PARA 9 "" 1 "" {TEXT -1 104 "\{--> en ter collect, args = diff(diff(y(x),x),x)-y(x), \{y(x), diff(y(x),x), d iff(diff(y(x),x),x)\}, normal" }}{PARA 9 "" 1 "" {TEXT -1 78 "<-- exit collect (now in dsolve/inttrans/solveit) = diff(diff(y(x),x),x)-y(x) \}" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t2G,&-%%diffG6$-%\"yG6#%\"xG- %\"$G6$F,\"\"#\"\"\"F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t2G7 $-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F,\"\"#,$F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%tlcmG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t3 G-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F+\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t3G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t3G,$-%\"yG6# %\"xG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t3G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&neqnsG,&-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F, \"\"#\"\"\"F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(ckicondG<\"" }}{PARA 9 "" 1 "" {TEXT -1 72 "\{--> enter inttrans[laplace], args = \+ \{diff(diff(y(x),x),x)-y(x)\}, x, _s1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$optG%$INTG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solnG<#,(*&%$_s 1G\"\"\",&*&F(F)-%(laplaceG6%-%\"yG6#%\"xGF2F(F)F)-F06#\"\"!!\"\"F)F)- -%\"DG6#F0F4F6F,F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+transformsG< \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solnG<#,(*&%$_s1G\"\"\",&*&F( F)-%(laplaceG6%-%\"yG6#%\"xGF2F(F)F)-F06#\"\"!!\"\"F)F)--%\"DG6#F0F4F6 F,F6" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>-&%)inttransG6#%(laplaceG6%<# ,&-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F2\"\"#\"\"\"F/!\"\"F2%$_s1G<#,(*&F9F 7,&*&F9F7-F(6%F/F2F9F7F7-F06#\"\"!F8F7F7--%\"DG6#F0FBF8F?F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#,(*&%$_s1G\"\"\",&*&F&F'-%(laplaceG6%-%\"y G6#%\"xGF0F&F'F'-F.6#\"\"!!\"\"F'F'--%\"DG6#F.F2F4F*F4" }}{PARA 9 "" 1 "" {TEXT -1 127 "<-- exit inttrans[laplace] (now in dsolve/inttrans/ solveit) = \{_s1*(_s1*laplace(y(x),x,_s1)-y(0))-D(y)(0)-laplace(y(x),x ,_s1)\}\}" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&transG<#,(*&%$_s1G\"\" \",&*&F(F)-%(laplaceG6%-%\"yG6#%\"xGF2F(F)F)-F06#\"\"!!\"\"F)F)--%\"DG 6#F0F4F6F,F6" }}{PARA 9 "" 1 "" {TEXT -1 51 "\{--> enter inttrans[lapl ace], args = \{y(x)\}, x, _s1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$op tG%$INTG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solnG<#-%(laplaceG6%-% \"yG6#%\"xGF,%$_s1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+transformsG< \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solnG<#-%(laplaceG6%-%\"yG6#% \"xGF,%$_s1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-&%)inttransG6#%(lapl aceG6%<#-%\"yG6#%\"xGF.%$_s1G<#-F(6%F+F.F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#-%(laplaceG6%-%\"yG6#%\"xGF*%$_s1G" }}{PARA 9 "" 1 " " {TEXT -1 84 "<-- exit inttrans[laplace] (now in dsolve/inttrans/solv eit) = \{laplace(y(x),x,_s1)\}\}" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% 'trans1G<#-%(laplaceG6%-%\"yG6#%\"xGF,%$_s1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&transG<#,(*&%$_s1G\"\"\",&*&F(F)-%1laplace/internalG 6%-%\"yG6#%\"xGF2F(F)F)-F06#\"\"!!\"\"F)F)--%\"DG6#F0F4F6F,F6" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'trans1G<#-%1laplace/internalG6%-%\" yG6#%\"xGF,%$_s1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(initvalG<$/-% \"yG6#\"\"!\"\"\"/--%\"DG6#F(F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"iG/-%\"yG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"iG/-- %\"DG6#%\"yG6#\"\"!F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&transG<#,& *&%$_s1G\"\"\",&*&F(F)-%1laplace/internalG6%-%\"yG6#%\"xGF2F(F)F)F)!\" \"F)F)F,F3" }}{PARA 9 "" 1 "" {TEXT -1 132 "\{--> enter solve, args = \+ \{_s1^2*`laplace/internal`(y(x),x,_s1)-_s1-`laplace/internal`(y(x),x,_ s1)\}, \{`laplace/internal`(y(x),x,_s1)\}" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%ArgsG7$<#,(*&)%$_s1G\"\"#\"\"\"-%1laplace/internalG6 %-%\"yG6#%\"xGF3F*F,F,F*!\"\"F-F4<#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,solve/splitG%&falseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(maxso lsG%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&tvarsG<#-%1laplac e/internalG6%-%\"yG6#%\"xGF,%$_s1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%3Solutions_are_setsG%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"t G<\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"tG<\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ROG<\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%ArgsG7$ <#,(*&)%$_s1G\"\"#\"\"\"-%1laplace/internalG6%-%\"yG6#%\"xGF3F*F,F,F*! \"\"F-F4<#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%4_SolutionsMayBeLost GF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%3solve/rec/rememberGF$" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%*solve/recGf*6%%&oeqnsG%&ineqsG%%var sG6.%%eqnsG%&eqns2G%'ineqs2G%&vars2G%)subspatrG%\"zG%\"rG%#nvG%&eqns1G %'ineqs1G%\"iG%'labelsG6$%)rememberG%ipCopyright~(c)~1993~Gaston~Gonne t,~Wissenschaftliches~Rechnen,~ETH~Zurich.~All~rights~reserved.G6\"C.> 8$-%'removeG6$f*6#%\"xGF:6$%)operatorG%&arrowGF:/-%&evalbG6#/9$\"\"!%% trueGF:F:F:FL@(-%$hasG6$F=%%FAILGO7\"5/F=<\"/F=<#FM@%-%'memberG6$FN-%$ mapG6$%'testeqG9%OFUO7#FX4-%%typeG6$9&-%$setG6#%%nameGC(>8%F=>8&F]o>8' Feo>8(%%NULLG?&8)FeoFN@$4-Fco6$FepFioC(?&8*-%'indetsG6$-%&unionG6$F\\p F^p-%(anyfuncG6$%*algebraicG%)anythingGFN@$3-FQ6$-%#opG6$\"\"\"F\\qFep -%#isG6$-F]r6$FMF\\q%*LinearMapGY6&Q)true,~%0F:QHcannot~solve~expressi ons~with~%1~for~%2F:F\\qFep>8+-%-solve/newvarGF:>F\\p-%%subsG6$/FepF[s F\\p>F^p-F`s6$FbsF^p>F`p-F`s6$FbsF`p>Fbp6$Fbp/F[sFepO-F`s6$Fbp-F$6%F\\ pF^pF`p>8,F=>8-F]o?(8.F_rF_r-%%nopsG6#FeoFNC&>Fbt-%*traperrorG6#-F`s6$ /&Feo6#Fft(%,solve/dummyGFftFbt@$/Fbt%*lasterrorG[>Fdt-F]u6#-F`s6$FauF dt@$/FdtFhuFiu>8/-F^q6$<$FbtFdt-%)specfuncG6$Fgq.%'RootOfG>Fbv-%'selec tG6%Fco-Fjn6$F]rFbv/-%*identicalG6#.%&labelGFgq>6$FbtFdt-F]r6#-F`s6$-% $seqG6$/&FbvFcu/Few-%$catG6$%2solve/dummy/labelGFft/Fft;F_r-Fht6#Fbv7$ FbtFdt@$30FbtFhu0FdtFhuC$@$/&%3solve/rec/rememberGFhwF_rC%@$33/FgtF_r/ 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OFgjl@$FeimC&>Fgen-Fb[m6%Fd\\l-F\\z6$F]jmF^coF\\u>8;-Fay6$f*F^eoFGF`eo FG/FP-Fial6$FPFbxFGFGFGFgen>8<-Fay6$f*F^eoFGF`eoFG/FccpFPFGFGFGFgen@$- Fdq6$-Fiq6$-FN6$F^cp<#/^#F`r.-Fcx6#,&*$)FhjnFjnF`rF`rF`rF`rF]jm-Fi\\l6 #<$F_]m%'algnumGC&>Fb`nFcw>%-_EnvExplicitGFj]n?&F`cl-Fe`n6$-Fiq6$-FN6$ <#Fi`nF^cpF]jmFbepFfqC$@$F]anFbem>Fb`n6$Fb`n<#/F\\`n-Fiq6$FfcpF`clOFha n7#-%-solve/gensysG6&F\\qF\\uF^qFddmFG6%%4_SolutionsMayBeLostG%3solve/ rec/rememberGFcelFG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+_EnvFloatsG% &falseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%A_EnvSimplifyRootOf_DoNot MultiplyG%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)TestzeroGf*6#% \"xG6\"6$%)operatorG%&arrowGF(-%&evalbG6#/-%'normalG6#-_%+SolveToolsG% /CancelInversesG6#9$\"\"!F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%3 Solutions_are_setsG%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0_EnvS ystemOfEqsG%&falseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG7#<#/-%1l aplace/internalG6%-%\"yG6#%\"xGF.%$_s1G*&F/\"\"\",&*$)F/\"\"#F1F1F1!\" \"F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG7#<#/-%1laplace/internal G6%-%\"yG6#%\"xGF.%$_s1G*&F/\"\"\",&*$)F/\"\"#F1F1F1!\"\"F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG7#<#/-%1laplace/internalG6%-%\"yG6#%\" xGF.%$_s1G*&F/\"\"\",&*$)F/\"\"#F1F1F1!\"\"F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG7#<#/-%1laplace/internalG6%-%\"yG6#%\"xGF.%$_s1G* &F/\"\"\",&*$)F/\"\"#F1F1F1!\"\"F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"rG7#<#/-%1laplace/internalG6%-%\"yG6#%\"xGF.%$_s1G*&F/\"\"\",&*$)F /\"\"#F1F1F1!\"\"F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG7#<#/-%1l aplace/internalG6%-%\"yG6#%\"xGF.%$_s1G*&F/\"\"\",&*$)F/\"\"#F1F1F1!\" \"F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG7#<#/-%1laplace/internal G6%-%\"yG6#%\"xGF.%$_s1G*&F/\"\"\",&*$)F/\"\"#F1F1F1!\"\"F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG7#<#/-%1laplace/internalG6%-%\"yG6#%\" xGF.%$_s1G*&F/\"\"\",&*$)F/\"\"#F1F1F1!\"\"F6" }}{PARA 9 "" 1 "" {TEXT -1 99 "<-- exit solve (now in dsolve/inttrans/solveit) = \{`lapl ace/internal`(y(x),x,_s1) = _s1/(_s1^2-1)\}\}" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&transG<#/-%1laplace/internalG6%-%\"yG6#%\"xGF-%$_s1G *&F.\"\"\",&*$)F.\"\"#F0F0F0!\"\"F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%%sollG/-%1laplace/internalG6%-%\"yG6#%\"xGF,%$_s1G*&F-\"\"\",&*$)F- \"\"#F/F/F/!\"\"F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sollG*&%$_s1G \"\"\",&*$)F&\"\"#F'F'F'!\"\"F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%' trans1G<#/-%1laplace/internalG6%-%\"yG6#%\"xGF-%$_s1G*&F.\"\"\",&*$)F. \"\"#F0F0F0!\"\"F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'trans1G<#/-% \"yG6#%\"xG-%%coshGF)" }}{PARA 9 "" 1 "" {TEXT -1 75 "<-- exit dsolve/ inttrans/solveit (now in dsolve/inttrans) = y(x) = cosh(x)\}" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&solnsG<#/-%\"yG6#%\"xG-%%coshGF)" }} {PARA 9 "" 1 "" {TEXT -1 61 "\{--> enter dsolve/closed_form/consts, ar gs = \{y(x) = cosh(x)\}" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&nexprG<# /-%\"yG6#%\"xG-%%coshGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"jG\"\" !" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#/-%\"yG6#%\"xG-%%coshGF'" }} {PARA 9 "" 1 "" {TEXT -1 79 "<-- exit dsolve/closed_form/consts (now i n dsolve/inttrans) = \{y(x) = cosh(x)\}\}" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&solnsG<#/-%\"yG6#%\"xG-%%coshGF)" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"yG6#%\"xG-%%coshGF&" }}{PARA 9 "" 1 "" {TEXT -1 67 "<-- exit dsolve/inttrans (now in dsolve/INTTRANS) = y(x) = cosh(x) \}" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%%coshGF&" }} {PARA 9 "" 1 "" {TEXT -1 58 "<-- exit dsolve/INTTRANS (now in dsolve) \+ = y(x) = cosh(x)\}" }}{PARA 9 "" 1 "" {TEXT -1 52 "<-- exit dsolve (no w at top level) = y(x) = cosh(x)\}" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%\"yG6#%\"xG-%%coshGF&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 288 2 "6." }{TEXT -1 141 " Comput e the first 10 terms in a Taylor series solution of the following init ial value problem.\n " } {XPPEDIT 18 0 "`y'` = y*z" "6#/%#y'G*&%\"yG\"\"\"%\"zGF'" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "`z'` = x*z+y" "6#/%#z'G,&*&%\"xG\"\"\"%\"zGF(F( %\"yGF(" }{TEXT 267 5 ", y" }{TEXT -1 3 "(0)" }{TEXT 269 3 " = " } {TEXT -1 1 "1" }{TEXT 270 5 ", z" }{TEXT -1 3 "(0)" }{TEXT 268 3 " = " }{TEXT -1 2 "0\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restar t:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "eqns := diff(y(x),x)= y(x)*z(x),diff(z(x),x)=x*z(x)+y(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%%eqnsG6$/-%%diffG6$-%\"yG6#%\"xGF-*&F*\"\"\"-%\"zGF,F//-F(6$F0F-,&* &F-F/F0F/F/F*F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "initvals := y(0)=1, z(0)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)initvalsG6$/ -%\"yG6#\"\"!\"\"\"/-%\"zGF)F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Order := 10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&OrderG\"#5 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "dsolve(\{eqns, initvals \}, \{y(x),z(x)\}, type=series);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$ /-%\"yG6#%\"xG+/F(\"\"\"\"\"!#F*\"\"#F-#F*\"\"%F/#\"#8\"$?\"\"\"'#\"#H \"$s'\"\")-%\"OG6#F*\"#5/-%\"zGF'+/F(F*F*F,\"\"$#F@\"#?\"\"&#\"#J\"$S) \"\"(#\"$p#\"&S-$\"\"*F8F;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 289 2 "7." }{TEXT -1 104 " Cons ider Airy's differential equation,\n \+ " }{XPPEDIT 18 0 "`y'' `+x*y=0" "6#/,&%% y''~G\"\"\"*&%\"xGF&%\"yGF&F&\"\"!" }{TEXT -1 2 ".\n" }{TEXT 290 3 "(a )" }{TEXT -1 38 " Find the solution for initial values " }{XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "`y'`(0)=0" "6#/-%#y'G6#\"\"!F'" }{TEXT -1 88 " via the power series method. Which terms of degree less than 30 occur in the solution?\n" }{TEXT 291 3 "(b)" }{TEXT -1 87 " What recurrence relation holds for t he coefficients of the power series found in (a)?\n" }{TEXT 292 3 "(c) " }{TEXT -1 38 " Find the solution for initial values " }{XPPEDIT 18 0 "y(0)=0" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "`y '`(0)=1" "6#/-%#y'G6#\"\"!\"\"\"" }{TEXT -1 88 " via the power series \+ method. Which terms of degree less than 30 occur in the solution?\n" } }{SECT 1 {PARA 0 "" 0 "" {TEXT 298 3 "(a)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Airy := diff(y(x),x$2) + x*y(x) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%AiryG/,&-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&F-F2F*F2F2 \"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "initvals := y(0)=1 , D(y)(0)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)initvalsG6$/-%\"yG6 #\"\"!\"\"\"/--%\"DG6#F(F)F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(powseries):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "s olution := powsolve(\{Airy, initvals\}):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "tpsform( solution, x, 30);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+9%\"xG\"\"\"\"\"!#!\"\"\"\"'\"\"$#F%\"$!=F)#F(\"&gH\" \"\"*#F%\"(?2r\"\"#7#F(\"*+7Df$\"#:#F%\"-+s'3$*4\"\"#=#F(\"/+SAk4 " 0 "" {MPLTEXT 1 0 30 "dsolve(\{Airy,initvals\}, y(x));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&#\"\"\"\"\"#F+*(-%& GAMMAG6##F,\"\"$F+)F2F1F+-%'AiryAiG6#,$F'!\"\"F+F+F+*&F*F+*(F.F+)F2#F+ \"\"'F+-%'AiryBiGF6F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*& #\"\"\"\"\"#F+*&,&*&)\"\"$#F,F1F+-%'AiryAiG6#,$F'!\"\"F+F+*&)F1#F+\"\" 'F+-%'AiryBiGF5F+F+F+-%&GAMMAG6#F2F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "convert(%,Bessel);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 #/-%\"yG6#%\"xG,$*&#\"\"\"\"\"#F+*&,&*&#F+\"\"$F+*(F1#F,F1,&*&F'F+-%(B esselIG6$F0,$*(F,F+F1!\"\",$*$)F'F1F+F;F*F+F+F+*&-F76$#F;F1F9F+)F " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*&#\"\"\"\"\"$F+**)F,#\"\"#F ,F+-%(BesselIG6$#!\"\"F,,$*(F0F+F,F5,$*$)F'F,F+F5#F+F0F+F+)F8#F+\"\"'F +-%&GAMMAG6#F/F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "ser ies(rhs(%), x, 30);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+9%\"xG\"\"\"\" \"!#!\"\"\"\"'\"\"$#F%\"$!=F)#F(\"&gH\"\"\"*#F%\"(?2r\"\"#7#F(\"*+7Df$ \"#:#F%\"-+s'3$*4\"\"#=#F(\"/+SAk4 " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 299 3 "(b)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Airy := diff(y(x),x$2) + x*y(x) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%AiryG/,&-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"# \"\"\"*&F-F2F*F2F2\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 " initvals := y(0)=1, D(y)(0)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)i nitvalsG6$/-%\"yG6#\"\"!\"\"\"/--%\"DG6#F(F)F*" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 16 "with(powseries):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "solution := powsolve(\{Airy, initvals\}):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "solution(_k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(-%\"aG6#,&%#_kG\"\"\"\"\"$!\"\"F*F)F,,&F)F*F*F,F,F, " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 70 "So, the recursion relation for the coefficients in the po wer series is" }}{EXCHG {PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "k*(k-1)*a[k]+a[k-3]=0" "6#/,&*(%\"kG\"\"\",&F&F'F'!\"\"F'&%\"aG6 #F&F'F'&F+6#,&F&F'\"\"$F)F'\"\"!" }{TEXT -1 1 " " }{MPLTEXT 1 0 0 "" } }}}{SECT 1 {PARA 0 "" 0 "" {TEXT 300 3 "(c)" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Airy := diff(y(x),x$2) + x*y(x) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%AiryG/,&-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&F-F2F*F 2F2\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "initvals := y(0 )=0, D(y)(0)=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)initvalsG6$/-%\" yG6#\"\"!F*/--%\"DG6#F(F)\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(powseries):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "solution := powsolve(\{Airy, initvals\}):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 26 "tpsform( solution, x, 30);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+9%\"xG\"\"\"F%#!\"\"\"#7\"\"%#F%\"$/&\"\"(#F'\"&g`%\"# 5#F%\"(gh2(\"#8#F'\"++%y#)p\"\"#;#F%\"-+G@63e\"#>#F'\"0+OJ!yM$o#\"#A#F %\"3++;)=o3+h\"\"#D#F'\"6++'*[-Ncmr@\"\"#G-%\"OG6#F%\"#I" }}}{PARA 0 " " 0 "" {TEXT -1 32 "Compare with the exact solution:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "dsolve(\{Airy,initvals\}, y(x));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&#\"\"\"\"\"$F+**%#Pi GF+-%&GAMMAG6##\"\"#F,!\"\"F,#\"\"&\"\"'-%'AiryAiG6#,$F'F4F+F+F+*&#F+F ,F+**F.F+F/F4F,F*-%'AiryBiGF:F+F+F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\" yG6#%\"xG,$*&#\"\"\"\"\"$F+*(%#PiGF+,&*&)F,#\"\"&\"\"'F+-%'AiryAiG6#,$ F'!\"\"F+F+*&)F,F*F+-%'AiryBiGF7F+F9F+-%&GAMMAG6##\"\"#F,F9F+F+" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "series(rhs(%), x, 30);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#+9%\"xG\"\"\"F%#!\"\"\"#7\"\"%#F%\"$/& \"\"(#F'\"&g`%\"#5#F%\"(gh2(\"#8#F'\"++%y#)p\"\"#;#F%\"-+G@63e\"#>#F' \"0+OJ!yM$o#\"#A#F%\"3++;)=o3+h\"\"#D#F'\"6++'*[-Ncmr@\"\"#G-%\"OG6#F% \"#I" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 293 2 "8." }{TEXT -1 110 " Consider Duffing's di fferential equation,\n \+ " }{XPPEDIT 18 0 "`x''`+x+epsilon *x^3=epsilon*F*co s*omega*t" "6#/,(%$x''G\"\"\"%\"xGF&*&%(epsilonGF&*$F'\"\"$F&F&*,F)F&% \"FGF&%$cosGF&%&omegaGF&%\"tGF&" }{TEXT -1 199 ".\nApply the Poincar \351-Lindstedt method to find an approximation of a priodic solution o f the ODE that satisfies the initial values\n \+ " }{XPPEDIT 18 0 "x(0)=A" "6#/-%\"xG6#\"\"!%\"AG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "`x'`(0)=0 " "6#/-%#x'G6#\"\"!F'" }{TEXT -1 130 " \nWhat are the results when you apply the method of multiple scales and how do both methods compare t o a numerical approximation?\n" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 30 "The Poincar\351-Lindstedt method " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "diff(x(t), t$2) + x(t) + epsilon*x(t)^3 - epsilon*F*cos(omega*t) = 0; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,*-%%diffG6$-%\"xG6#%\"tG-%\"$G6 $F+\"\"#\"\"\"F(F0*&%(epsilonGF0)F(\"\"$F0F0*(F2F0%\"FGF0-%$cosG6#*&%& omegaGF0F+F0F0!\"\"\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 39 "We stretch t ime via the transformation " }{XPPEDIT 18 0 "tau=omega*t" "6#/%$tauG*& %&omegaG\"\"\"%\"tGF'" }{TEXT -1 35 ". The differential equation becom es" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "ODE := DEtools[Dchange var]( \n \{ t=tau/omega, x(t)=x(tau) \}, %, t, tau );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ODEG/,**&)%&omegaG\"\"#\"\"\"-%%diffG6$-%\"xG6# %$tauG-%\"$G6$F2F*F+F+F/F+*&%(epsilonGF+)F/\"\"$F+F+*(F7F+%\"FGF+-%$co sGF1F+!\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "e_order := 6: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "macro(e=epsilon, t=tau):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "alias(seq(x[i] \+ = xi[i](tau), i=0..e_order)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "e := () -> e: # introduce e as a constant function " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "for i from 0 to e_order do \n xi [i] := t -> xi[i](t)\nend do: " }}}{PARA 0 "" 0 "" {TEXT -1 10 "We ass ume " }{XPPEDIT 18 0 "omega=1+omega[1]*epsilon+omega[2]*epsilon^2+`... `" "6#/%&omegaG,*\"\"\"F&*&&F$6#F&F&%(epsilonGF&F&*&&F$6#\"\"#F&*$F*F. F&F&%$...GF&" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x(tau)=x[0](tau)+x[1 ](tau)*epsilon+x[2](tau)*epsilon^2+`...`" "6#/-%\"xG6#%$tauG,*-&F%6#\" \"!6#F'\"\"\"*&-&F%6#F.6#F'F.%(epsilonGF.F.*&-&F%6#\"\"#6#F'F.*$F4F9F. F.%$...GF." }{TEXT -1 61 " and determine the differential equations of these functions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "omega : = 1 + sum( 'w[i]*e^i', 'i'=1..e_order );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&omegaG,0\"\"\"F&*&&%\"wG6#F&F&%(epsilonGF&F&*&&F)6#\"\"#F&)F+ F/F&F&*&&F)6#\"\"$F&)F+F4F&F&*&&F)6#\"\"%F&)F+F9F&F&*&&F)6#\"\"&F&)F+F >F&F&*&&F)6#\"\"'F&)F+FCF&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "x := sum('xi[i]*e^i', 'i'=0..e_order);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG,0&%#xiG6#\"\"!\"\"\"*&&F'6#F*F*%(epsilonGF*F**&& F'6#\"\"#F*)F.F2F*F**&&F'6#\"\"$F*)F.F7F*F**&&F'6#\"\"%F*)F.F " 0 " " {MPLTEXT 1 0 53 "deqn := simplify(collect(ODE,e), \{e^(e_order+1)=0 \}): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "for i from 0 to e_ order do\n ode[i] := coeff(lhs(deqn), e, i) = 0\nend do: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "ode[0];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%%diffG6$&%\"xG6#\"\"!-%\"$G6$%$tauG\"\"#\"\"\"F(F1 F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "ode[1];" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/,,-%%diffG6$&%\"xG6#\"\"\"-%\"$G6$%$tauG\"\"#F+ *(F0F+&%\"wGF*F+-F&6$&F)6#\"\"!F,F+F+F(F+*&%\"FGF+-%$cosG6#F/F+!\"\"*$ )F6\"\"$F+F+F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "ode[2];" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/,.&%\"xG6#\"\"#\"\"\"-%%diffG6$F%-% \"$G6$%$tauGF(F)*(F(F)&%\"wG6#F)F)-F+6$&F&F4F-F)F)*(F(F)-F+6$&F&6#\"\" !F-F)&F3F'F)F)*&F9F))F2F(F)F)*(\"\"$F))F;F(F)F7F)F)F=" }}}{PARA 0 "" 0 "" {TEXT -1 19 "The initial values " }{XPPEDIT 18 0 "x(0)=A " "6#/-% \"xG6#\"\"!%\"AG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "`x'`(0)=0" "6#/- %#x'G6#\"\"!F'" }{TEXT -1 16 " translate into " }{XPPEDIT 18 0 "x[0](0 )=A, x[1](0)=0, x[2](0)=0,`...`" "6&/-&%\"xG6#\"\"!6#F(%\"AG/-&F&6#\" \"\"6#F(F(/-&F&6#\"\"#6#F(F(%$...G" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "`x'`[0](0)=0, `x'`[1](0)=0,`x'`[2](0)=0, `...`" "6&/-&%#x'G6#\"\"!6 #F(F(/-&F&6#\"\"\"6#F(F(/-&F&6#\"\"#6#F(F(%$...G" }{TEXT -1 2 " ." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "dsolve(\{ode[0], xi[0](0)=A, D(xi[0])(0)=0\}, " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " xi[0](t), m ethod=laplace);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"xG6#\"\"!*&%\" AG\"\"\"-%$cosG6#%$tauGF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "xi[0] := unapply(rhs(%), t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>& %#xiG6#\"\"!f*6#%$tauG6\"6$%)operatorG%&arrowGF+*&%\"AG\"\"\"-%$cosG6# 9$F1F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "ode[1];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,,-%%diffG6$&%\"xG6#\"\"\"-%\"$G6$%$t auG\"\"#F+**F0F+&%\"wGF*F+%\"AGF+-%$cosG6#F/F+!\"\"F(F+*&%\"FGF+F5F+F8 *&)F4\"\"$F+)F5F=F+F+\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "map(combine, ode[1], 'trig'); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/,.-%%diffG6$&%\"xG6#\"\"\"-%\"$G6$%$tauG\"\"#F+**F0F+&%\"wGF*F+%\"AG F+-%$cosG6#F/F+!\"\"F(F+*&%\"FGF+F5F+F8*&#F+\"\"%F+*&)F4\"\"$F+-F66#,$ *&F@F+F/F+F+F+F+F+*&#F@F=F+*&F?F+F5F+F+F+\"\"!" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 36 "ode[1] := map(collect, %, [cos(t)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$odeG6#\"\"\"/,**&,(*(\"\"#F'&%\"wGF&F'% \"AGF'!\"\"%\"FGF1*(\"\"$F'\"\"%F1F0F4F'F'-%$cosG6#%$tauGF'F'-%%diffG6 $&%\"xGF&-%\"$G6$F9F-F'F=F'*&#F'F5F'*&)F0F4F'-F76#,$*&F4F'F9F'F'F'F'F' \"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 33 "We let vanish the resonance ter m " }{XPPEDIT 18 0 "cos(tau)" "6#-%$cosG6#%$tauG" }{TEXT -1 1 "." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "solve(\{coeff( lhs(ode[1]), \+ cos(t)) = 0\}, w[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#/&%\"wG6#\" \"\",$*(\"\")!\"\",&*&\"\"%F(%\"FGF(F(*&\"\"$F()%\"AGF2F(F,F(F4F,F," } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "assign(%);" }}}{PARA 0 "" 0 "" {TEXT -1 38 "We solve the differential equation of " }{XPPEDIT 18 0 "x[1](tau)" "6#-&%\"xG6#\"\"\"6#%$tauG" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "ode[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(-%%diffG6$&%\"xG6#\"\"\"-%\"$G6$%$tauG\"\"#F+F(F+*&# F+\"\"%F+*&)%\"AG\"\"$F+-%$cosG6#,$*&F7F+F/F+F+F+F+F+\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "dsolve(\{ode[1], xi[1](0)=0, D(xi[1 ])(0)=0\}, " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " xi[1](t), method=l aplace);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"xG6#\"\"\"*&)%\"AG\" \"$F',&*&#F'\"#KF'-%$cosG6#,$*&F+F'%$tauGF'F'F'F'*&#F'F/F'-F16#F5F'!\" \"F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "xi[1] := unapply(rh s(%), tau); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#xiG6#\"\"\"f*6#%$t auG6\"6$%)operatorG%&arrowGF+*&)%\"AG\"\"$F',&*&#F'\"#KF'-%$cosG6#,$*& F2F'9$F'F'F'F'*&#F'F6F'-F86#F " 0 "" {MPLTEXT 1 0 30 "map(combine, ode[2], 'trig'): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "ode[2] := map(collect, %, [cos(t), cos(3* t)]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>&%$odeG6#\"\"#/,(*&#\"\"\"\" $G\"F,*(,**(\"#WF,%\"FGF,)%\"AG\"\"$F,F,*&\"#@F,)F4\"\"'F,!\"\"*(\"$c# F,)F4F'F,&%\"wGF&F,F:*&\"#KF,)F2F'F,F:F,F4F:-%$cosG6#%$tauGF,F,F,*&F+F ,*(,&*(\"#OF,F2F,F3F,F,*&\"#CF,F8F,F:F,F4F:-FD6#,$*&F5F,FFF,F,F,F,F,*& F+F,*&,(*(F-F,&%\"xGF&F,F4F,F,*(F-F,-%%diffG6$FV-%\"$G6$FFF'F,F4F,F,*( F5F,F8F,-FD6#,$*&\"\"&F,FFF,F,F,F,F,F4F:F,F,\"\"!" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 47 "solve(\{coeff(lhs(ode[2]), cos(t)) = 0\}, w[ 2]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#/&%\"wG6#\"\"#,$*(\"$c#!\" \",(*(\"#W\"\"\"%\"FGF0)%\"AG\"\"$F0F,*&\"#@F0)F3\"\"'F0F0*&\"#KF0)F1F (F0F0F0F3!\"#F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "assign(% ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "dsolve(\{ode[2], xi[2 ](0)=0, D(xi[2])(0)=0\}, " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " xi[2 ](t), method=laplace):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "c ollect(%, [cos(t), cos(3*t), cos(5*t)]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "xi[2] := unapply(rhs(%), t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#xiG6#\"\"#f*6#%$tauG6\"6$%)operatorG%&arrowGF+,(*() %\"AGF'\"\"\",&*(\"#BF3\"%C5!\"\"F2\"\"$F3*(\"\"*F3\"$c#F8%\"FGF3F8F3- %$cosG6#9$F3F3*(F1F3,&*(F;F3F " 0 "" {MPLTEXT 1 0 379 "for i from 3 to e_order do \+ \n map(combine, ode[i], 'trig'):\n ode[i] := map(collect, %, \n [ seq(cos((2*j+1)*t),j=0..i)]):\n solve(\{coeff(lhs(ode[i]), cos(t)) = \+ 0\}, w[i]):\n assign(%):\n dsolve(\{ode[i], xi[i](0)=0, D(xi[i])(0)= 0\},\n xi[i](t), method=laplace):\n collect(%, [seq(sin((2*j+1)*t) ,j=0..i), \n seq(cos((2*j+1)*t),j=0..i)]):\n xi[i] := unapply(rhs( %), t)\nend do: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "omega; \+ " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,0\"\"\"F$**\"\")!\"\",&*&\"\"%F$ %\"FGF$F$*&\"\"$F$)%\"AGF-F$F'F$F/F'%(epsilonGF$F'**\"$c#F',(*(\"#WF$F +F$F.F$F'*&\"#@F$)F/\"\"'F$F$*&\"#KF$)F+\"\"#F$F$F$F/!\"#F0F=F'**\"%[? F',**(\"$O#F$FF$)F/\"#7F$F $F$F/!\"%F0F*F'**\")oV()=F',.*(\"(Z:X\"F$F+F$FfnF$F$*(\"(cA6\"F$FYF$F. F$F'*&\"''4;&F$)F+\"\"&F$F$*&\"'L'R$F$)F/\"#:F$F'*(\"(1gP#F$FF$FLF$F8F$F$F$F/!\"&F0FcoF'**\"+Smpf!*F',0*(\"*g@*GNF$FboF$F. F$F'*&\"*gXz&=F$)F+F9F$F$*&\"*0ZQE\"F$)F/\"#=F$F$*(\"**R*=a'F$F+F$FfoF $F'*(\"*+smG(F$FYF$F8F$F$*(\"+g#\\!Q8F$FLF$FFF$F'*(\"+Mp$\\L\"F$F " 0 "" {MPLTEXT 1 0 7 "x(t); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,0*&%\"AG\"\"\"-%$cosG6#%$tauGF& F&*()F%\"\"$F&,&*&#F&\"#KF&-F(6#,$*&F-F&F*F&F&F&F&*&#F&F1F&F'F&!\"\"F& %(epsilonGF&F&*&,(*()F%\"\"#F&,&*(\"#BF&\"%C5F8F%F-F&*(\"\"*F&\"$c#F8% \"FGF&F8F&F'F&F&*(F=F&,&*(FDF&FEF8FFF&F&*(F-F&\"$G\"F8F%F-F8F&F2F&F&*& #F&FBF&*&)F%\"\"&F&-F(6#,$*&FPF&F*F&F&F&F&F&F&)F9F>F&F&*&,**(F%F&,(*&# \"$Z&\"&oF$F&*$)F%\"\"'F&F&F8*&#\"%<7\"&wX#F&*&FFF&F,F&F&F&*&#\"#\")\" %[?F&*$)FFF>F&F&F8F&F'F&F&*(F%F&,(*&#FboFcoF&FdoF&F&*&#\"$B%\"%#>)F&F_ oF&F8*&#\"$(H\"&%Q;F&FhnF&F&F&F2F&F&*&#F&\"%WhF&*()F%\"\"%F&,&*&\"#8F& FFF&F&*&FDF&F,F&F8F&FQF&F&F&*&#F&FgnF&*&)F%\"\"(F&-F(6#,$*&F`qF&F*F&F& F&F&F&F&)F9F-F&F&*&,,*&#F&\"(w&[5F&*&)F%FDF&-F(6#,$*&FDF&F*F&F&F&F&F&* &,**&#\"$H(FapF&*$)FFF-F&F&F8*&#\"&0%G\"')GC&F&*&FFF&FinF&F&F8*&#\"%%F&*$)F%\"#7F&F&F8*&#\"(n'R))\")su\\vF&*&FeoF&Fin F&F&F8*&#\"'J0&)\"())y2(F&*&FgrF&F,F&F&F&*&#\"($4GM\")gX\"H'F&*&FFF&F \\rF&F&F&F&F%F8F'F&F&*&#F&\")KWbLF&*&)F%\"#6F&-F(6#,$*&F^xF&F*F&F&F&F& F&*&#F&\"*)))*)>IF&*(FOF&,(*(\"&$Q))F&FFF&F,F&F8*&\"&3T'F&FeoF&F&*&\"& E;$F&FinF&F&F&FaqF&F&F&*&#F-\"*g@xn\"F&*()F%\"\")F&,&*&\"$H#F&FFF&F&*& \"$g\"F&F,F&F8F&F]rF&F&F&*&#F&\"*K[)HXF&*(F=F&,**&\"(cID#F&FgrF&F&*(\" (t>A$F&FFF&FinF&F&*(\"(?([XF&FeoF&F,F&F8*&\"'WdzF&F\\rF&F8F&FQF&F&F&*( ,,*&#\"',ZRF[xF&FgvF&F&*&#F_vF`vF&FavF&F&*&#\"&Nm\"F`vF&F^wF&F&*&#\"&0 G$\"'W@EF&FcwF&F8*&#\"(\\'G5\");sx;F&FhwF&F8F&F2F&F%F8F&F&)F9FPF&F&*&, 0*(,.*&#\"*ZZR8#\"+S=`ESF&*&FFF&FhvF&F&F8*&#\"+jf.oV\",+)))*)>IF&*&Feo F&F\\rF&F&F&*&#\"(8*)4\"\"*Gx@M\"F&*$)F%\"#:F&F&F&*&#\"*R,=j(\"+c'yQi$ F&*&FgrF&FinF&F&F8*&#\")po;c\"*C'Q(R$F&*&FbvF&F,F&F&F&*&#\"&\\!fFjqF&* $)FFFPF&F&F8F&F%!\"#F'F&F&*&#F-\",+caVo#F&*()F%\"#5F&,&*&\"%H9F&FFF&F& *&\"%+5F&F,F&F8F&F_xF&F&F&*&#F&\",?JdxC(F&*(FfpF&,**&\"(+5!)*F&F\\rF&F 8*&\")gwjFF&FgrF&F&*(\")\"yh+%F&FFF&FinF&F&*(\")!3Fl&F&FeoF&F,F&F8F&Fa 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S7$$\"+z6LDAF]cl$!+#paE'**F17$$\"+r5PJAF]cl$!+O]K\\)*F17$$\"+i4TPAF]cl $!+Jq$)*p*F17$$\"+`05lAF]cl$!+KUB'f)F17$$\"+V,z#H#F]cl$!+Nf4`pF17$$\"+ Vgb)=&F17$$\"+U>KUBF]cl$!+gr^XKF17$$\"+cvsoBF]cl$!+z-]@5F1 7$$\"+qJ8&R#F]cl$\"+**>,i8F17$$\"+T*pxS#F]cl$\"+4pjhDF17$$\"+8nS?CF]cl $\"+=e'fz$F17$$\"+%[VIV#F]cl$\"+'zBf0&F17$$\"+b-oXCF]cl$\"+3LoCjF17$$ \"+#>g#fCF]cl$\"+G`RnwF17$$\"+G,%GZ#F]cl$\"+R\\cU*)F17$$\"+k+U'[#F]cl$ \"+\"*o'*35FS7$\"#D$\"+O7!R5\"FS-%+AXESLABELSG6$Q!6\"Fgan-%'COLOURG6&% $RGBGF(F(F(-%%VIEWG6$;$F(F($FaanF(%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 63 "In good agreement \+ except for the amplitude of the oscillations." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 25 "Method of Multiple Scales" }}{PARA 0 "" 0 "" {TEXT -1 63 "We shall apply the two-variable method to Duffing's equation: \+ " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "t[1]=t, t[2]=epsilon*t" "6$/&%\"tG6 #\"\"\"F%/&F%6#\"\"#*&%(epsilonGF'F%F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x(t) = x[0](t[1], t[2])+epsilon*x[1](t[1],t[2])" "6#/-%\"xG6#%\" tG,&-&F%6#\"\"!6$&F'6#\"\"\"&F'6#\"\"#F0*&%(epsilonGF0-&F%6#F06$&F'6#F 0&F'6#F3F0F0" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 44 "We deter mine the differential equations for " }{XPPEDIT 18 0 "x[0], x[1]" "6$& %\"xG6#\"\"!&F$6#\"\"\"" }{TEXT -1 23 ". We consider the case " } {XPPEDIT 18 0 "omega=1+epsilon*alpha" "6#/%&omegaG,&\"\"\"F&*&%(epsilo nGF&%&alphaGF&F&" }{TEXT -1 9 " so that " }{XPPEDIT 18 0 "omega*t=t[1] +alpha*t[2]" "6#/*&%&omegaG\"\"\"%\"tGF&,&&F'6#F&F&*&%&alphaGF&&F'6#\" \"#F&F&" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "r estart; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "alias(epsilon=e , \n seq(x[i] = xi[i](seq(t[j],j=1..2)), i=0..1)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "macro(t1=t[1], t2=t[2]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "e_order := 1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "e := subs( variables = seq( u||j, j=0..e_orde r), " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 " body=e, (variables -> bo dy) ): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "subs( D=sum(e^( i-1)*D[i], i=1..e_order+1)," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " (D @@e_order)(x) ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "simplif y(collect(%,e), \{e^(e_order+1)=0\}); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-&%\"DG6#\"\"\"6#%\"xGF(*&%(epsilonGF(-&F&6#\"\"#F)F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "convert(%(seq(t[j], j=1..e_o rder+1)), diff);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%%diffG6$-%\"xG 6$&%\"tG6#\"\"\"&F+6#\"\"#F*F-*&%(epsilonGF--F%6$F'F.F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Force := (t1,t2) -> F*cos(t1+alpha* t2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ForceGf*6$%#t1G%#t2G6\"6$%) operatorG%&arrowGF)*&%\"FG\"\"\"-%$cosG6#,&9$F/*&%&alphaGF/9%F/F/F/F)F )F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "ODE := (D@@2)(x) + x + epsilon*x^3 - epsilon*Force = 0; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ODEG/,*--%#@@G6$%\"DG\"\"#6#%\"xG\"\"\"F.F/*&%(epsilonGF/)F.\"\" $F/F/*&F1F/%&ForceGF/!\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "subs(D=sum('e^(i-1)*D[i]','i'=1..e_order+1), ODE):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "x := sum('xi[i]*e^i', 'i'=0..e_orde r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG,&&%#xiG6#\"\"!\"\"\"*&&F '6#F*F*%(epsilonGF*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "di ffeqn := simplify(collect(%%,e), \{e^(e_order+1)=0\}):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "for i from 0 to e_order do\n ode [i] := coeff(lhs(diffeqn), e, i) = 0\nend do; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$odeG6#\"\"!/,&-&%\"DG6$\"\"\"F.6#&%#xiGF&F.F0F.F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$odeG6#\"\"\"/,,-&%\"DG6$F'F'6#&% #xiGF&F'*&\"\"#F'-&F,6$F'F26#&F06#\"\"!F'F'*$)F7\"\"$F'F'F/F'%&ForceG! \"\"F9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "ode[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,,-&%\"DG6$\"\"\"F)6#&%#xiG6#F)F)*&\"\"#F)- &F'6$F)F/6#&F,6#\"\"!F)F)*$)F4\"\"$F)F)F+F)%&ForceG!\"\"F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "ode[1] := convert(ode[1](t1,t2), di ff);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$odeG6#\"\"\"/,,-%%diffG6$& %\"xGF&-%\"$G6$&%\"tGF&\"\"#F'*&F4F'-F+6%&F.6#\"\"!F2&F36#F4F'F'*$)F8 \"\"$F'F'F-F'*&%\"FGF'-%$cosG6#,&F2F'*&%&alphaGF'F;F'F'F'!\"\"F:" }}} {PARA 0 "" 0 "" {TEXT -1 38 "We consider the following solution of " } {XPPEDIT 18 0 "x[0]" "6#&%\"xG6#\"\"!" }{TEXT -1 50 " and subsitute it in the differential equation of " }{XPPEDIT 18 0 "x[1]" "6#&%\"xG6#\" \"\"" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "xi[0 ] := (t1,t2) -> A(t2)*cos(t1+B(t2));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>&%#xiG6#\"\"!f*6$%#t1G%#t2G6\"6$%)operatorG%&arrowGF,*&-%\"AG6#9%\" \"\"-%$cosG6#,&9$F5-%\"BGF3F5F5F,F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "ode[1];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,.-%%diffG 6$&%\"xG6#\"\"\"-%\"$G6$&%\"tGF*\"\"#F+*(F1F+-F&6$-%\"AG6#&F06#F1F8F+- %$sinG6#,&F/F+-%\"BGF7F+F+!\"\"**F1F+F5F+-%$cosGFF8F+F@*&)F5 \"\"$F+)FBFHF+F+F(F+*&%\"FGF+-FC6#,&F/F+*&%&alphaGF+F8F+F+F+F@\"\"!" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "ode[1] := combine(ode[1], \+ 'trig');" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>&%$odeG6#\"\"\"/,0-%%diff G6$&%\"xGF&-%\"$G6$&%\"tGF&\"\"#F'*(F4F'-F+6$-%\"AG6#&F36#F4F;F'-%$sin G6#,&F2F'-%\"BGF:F'F'!\"\"**F4F'F8F'-%$cosGF?F'-F+6$FAF;F'FC*&#F'\"\"% F'*&)F8\"\"$F'-FF6#,&*&FNF'F2F'F'*&FNF'FAF'F'F'F'F'*&#FNFKF'*&FMF'FEF' F'F'F-F'*&%\"FGF'-FF6#,&F2F'*&%&alphaGF'F;F'F'F'FC\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "subs(3*t1=Z, %);" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#/,0-%%diffG6$&%\"xG6#\"\"\"-%\"$G6$&%\"tGF*\"\"#F+*(F 1F+-F&6$-%\"AG6#&F06#F1F8F+-%$sinG6#,&F/F+-%\"BGF7F+F+!\"\"**F1F+F5F+- %$cosGFF8F+F@*&#F+\"\"%F+*&)F5\"\"$F+-FC6#,&%\"ZGF+*&FKF+F>F +F+F+F+F+*&#FKFHF+*&FJF+FBF+F+F+F(F+*&%\"FGF+-FC6#,&F/F+*&%&alphaGF+F8 F+F+F+F@\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(%); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,>-%%diffG6$&%\"xG6#\"\"\"-%\"$G6 $&%\"tGF*\"\"#F+**F1F+-F&6$-%\"AG6#&F06#F1F8F+-%$sinG6#F/F+-%$cosG6#-% \"BGF7F+!\"\"**F1F+F3F+-F>F6#%\"ZGF+)F=FLF+F+*&#FL\"\"%F+* (FKF+FMF+F=F+F+FB**FKF+-F;FNF+FEF+)F=F1F+FB*&#F+FSF+*(FKF+FVF+FEF+F+F+ *&#FLFSF+*(FKF+FDF+F=F+F+F+*&#FLFSF+*(FKF+F:F+FEF+F+FBF(F+*(%\"FGF+FDF +-F>6#*&%&alphaGF+F8F+F+FB*(F\\oF+F:F+-F;F^oF+F+\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "ode[1] := subs(Z=3*t1, %);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>&%$odeG6#\"\"\"/,>-%%diffG6$&%\"xGF&- %\"$G6$&%\"tGF&\"\"#F'**F4F'-F+6$-%\"AG6#&F36#F4F;F'-%$sinG6#F2F'-%$co sG6#-%\"BGF:F'!\"\"**F4F'F6F'-FAF?F'-F>FBF'FE*,F4F'F8F'-F+6$FCF;F'FGF' F@F'FE*,F4F'F8F'FJF'F=F'FHF'F'*()F8\"\"$F'-FA6#,$*&FOF'F2F'F'F')F@FOF' F'*&#FO\"\"%F'*(FNF'FPF'F@F'F'FE**FNF'-F>FQF'FHF')F@F4F'FE*&#F'FWF'*(F NF'FZF'FHF'F'F'*&#FOFWF'*(FNF'FGF'F@F'F'F'*&#FOFWF'*(FNF'F=F'FHF'F'FEF -F'*(%\"FGF'FGF'-FA6#*&%&alphaGF'F;F'F'FE*(F`oF'F=F'-F>FboF'F'\"\"!" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "ode[1] := collect(%, [sin( t1), cos(t1)]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>&%$odeG6#\"\"\"/,2 *&,**(\"\"#F'-%%diffG6$-%\"AG6#&%\"tG6#F-F4F'-%$cosG6#-%\"BGF3F'!\"\"* &#\"\"$\"\"%F'*&)F1F?F'-%$sinGF9F'F'F<**F-F'F1F'-F/6$F:F4F'FCF'F'*&%\" FGF'-FD6#*&%&alphaGF'F4F'F'F'F'-FD6#&F5F&F'F'*&,**(F-F'F.F'FCF'F<**F-F 'F1F'FFF'F7F'F<*&FIF'-F8FKF'F<*&#F?F@F'*&FBF'F7F'F'F'F'-F8FOF'F'-F/6$& %\"xGF&-%\"$G6$FPF-F'*(FBF'-F86#,$*&F?F'FPF'F'F')F7F?F'F'*&#F?F@F'*(FB F'F]oF'F7F'F'F<**FBF'-FDF^oF'FCF')F7F-F'F<*&#F'F@F'*(FBF'FfoF'FCF'F'F' FgnF'\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 50 "Eliminate the secular term s gives ristrictions on " }{XPPEDIT 18 0 "A(t[2])" "6#-%\"AG6#&%\"tG6# \"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "B(t[2])" "6#-%\"BG6#&%\"tG6 #\"\"#" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "re strictions := \{coeff(lhs(ode[1]), sin(t1))=0, \n coeff(lhs(ode[1]), \+ cos(t1))=0\};" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%-restrictionsG<$/,* *(\"\"#\"\"\"-%%diffG6$-%\"AG6#&%\"tG6#F)F1F*-%$cosG6#-%\"BGF0F*!\"\"* &#\"\"$\"\"%F**&)F.F " 0 "" {MPLTEXT 1 0 56 "restrictions[1]*cos(B(t2)) + restrictions[2] *sin(B(t2));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,&*&-%$cosG6#-%\"BG6# &%\"tG6#\"\"#\"\"\",**(F/F0-%%diffG6$-%\"AGF+F,F0F&F0!\"\"*&#\"\"$\"\" %F0*&)F6F;F0-%$sinGF(F0F0F8**F/F0F6F0-F46$F)F,F0F?F0F0*&%\"FGF0-F@6#*& %&alphaGF0F,F0F0F0F0F0*&F?F0,**(F/F0F3F0F?F0F8**F/F0F6F0FBF0F&F0F8*&FE F0-F'FGF0F8*&#F;FF0F&F0F0F0F0F0\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/, (*(-%$cosG6#-%\"BG6#&%\"tG6#\"\"#\"\"\"%\"FGF0-%$sinG6#*&%&alphaGF0F,F 0F0F0*&F/F0-%%diffG6$-%\"AGF+F,F0!\"\"*(-F3F(F0F1F0-F'F4F0F=\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "eq1 := combine(%, 'trig');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq1G/,&*&%\"FG\"\"\"-%$sinG6#,&-% \"BG6#&%\"tG6#\"\"#!\"\"*&%&alphaGF)F1F)F)F)F)*&F4F)-%%diffG6$-%\"AGF0 F1F)F5\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "restrictions [1]*sin(B(t2)) - restrictions[2]*cos(B(t2));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,&*&-%$sinG6#-%\"BG6#&%\"tG6#\"\"#\"\"\",**(F/F0-%%dif fG6$-%\"AGF+F,F0-%$cosGF(F0!\"\"*&#\"\"$\"\"%F0*&)F6F=F0F&F0F0F:**F/F0 F6F0-F46$F)F,F0F&F0F0*&%\"FGF0-F'6#*&%&alphaGF0F,F0F0F0F0F0*&F8F0,**(F /F0F3F0F&F0F:**F/F0F6F0FBF0F8F0F:*&FEF0-F9FGF0F:*&#F=F>F0*&F@F0F8F0F0F 0F0F:\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,**&#\"\"$\"\"%\"\"\"*$)-%\"AG6#& %\"tG6#\"\"#F'F)F)!\"\"*(F2F)F,F)-%%diffG6$-%\"BGF.F/F)F)*(-%$sinG6#F8 F)%\"FGF)-F<6#*&%&alphaGF)F/F)F)F)*(-%$cosGF=F)F>F)-FEF@F)F)\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "eq2 := combine(%, 'trig');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq2G/,(*&#\"\"$\"\"%\"\"\"*$)-%\" AG6#&%\"tG6#\"\"#F)F+F+!\"\"*(F4F+F.F+-%%diffG6$-%\"BGF0F1F+F+*&%\"FGF +-%$cosG6#,&F:F5*&%&alphaGF+F1F+F+F+F+\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 37 "So we get the following to equations:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "\{eq1,eq2\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$ /,&*&%\"FG\"\"\"-%$sinG6#,&-%\"BG6#&%\"tG6#\"\"#!\"\"*&%&alphaGF(F0F(F (F(F(*&F3F(-%%diffG6$-%\"AGF/F0F(F4\"\"!/,(*&#\"\"$\"\"%F(*$)F;FBF(F(F 4*(F3F(F;F(-F96$F-F0F(F(*&F'F(-%$cosGF+F(F(F=" }}}{PARA 0 "" 0 "" {TEXT -1 28 "Assuming that the amplitude " }{XPPEDIT 18 0 "A(t[2])" "6 #-%\"AG6#&%\"tG6#\"\"#" }{TEXT -1 13 " is constant " }{XPPEDIT 18 0 "A " "6#%\"AG" }{TEXT -1 38 " we have from the first equation that " } {XPPEDIT 18 0 "B(t[2])=alpha*t[2]" "6#/-%\"BG6#&%\"tG6#\"\"#*&%&alphaG \"\"\"&F(6#F*F-" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "eval(subs(\{A(t2)=A, B(t2)=alpha*t2\}, %));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#<$/\"\"!F%/,(*&#\"\"$\"\"%\"\"\"*$)%\"AGF*F,F,!\"\"*( \"\"#F,F/F,%&alphaGF,F,%\"FGF,F%" }}}{PARA 0 "" 0 "" {TEXT -1 50 "We c an solve the third degree polynomial equations" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "solve(%,A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#< #/%\"AG-%'RootOfG6#,(*&\"\"$\"\"\")%#_ZGF+F,F,*(\"\")F,F.F,%&alphaGF,! \"\"*&\"\"%F,%\"FGF,F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "c onvert(%,radical);" }}{PARA 11 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#<#/%\"AG,&*&\"\"$!\"\",&*&\"#=\"\"\"%\"FGF-F-*&\"\"#F -,&*&\"$G\"F-)%&alphaGF(F-F)*&\"#\")F-)F.F0F-F-#F-F0F-#F-F(F-**\"\")F- F(F)F5F-F*#F)F(F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "condA \+ := %:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "subs(\{A(t2)=rhs(o p(condA)), B(t2)=alpha*t2\}, xi[0](t1,t2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*&\"\"$!\"\",&*&\"#=\"\"\"%\"FGF+F+*&\"\"#F+,&*&\"$ G\"F+)%&alphaGF&F+F'*&\"#\")F+)F,F.F+F+#F+F.F+#F+F&F+**\"\")F+F&F'F3F+ F(#F'F&F+F+-%$cosG6#,&&%\"tG6#F+F+*&F3F+&FA6#F.F+F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 44 "We have obtained the following solution for " } {XPPEDIT 18 0 "x[0]" "6#&%\"xG6#\"\"!" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "subs(t1+alpha*t2=omega*t, %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*&\"\"$!\"\",&*&\"#=\"\"\"%\"FGF+F+*&\"\"#F+,&*&\"$ G\"F+)%&alphaGF&F+F'*&\"#\")F+)F,F.F+F+#F+F.F+#F+F&F+**\"\")F+F&F'F3F+ F(#F'F&F+F+-%$cosG6#*&%&omegaGF+%\"tGF+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "With the restrictions on " } {XPPEDIT 18 0 "A(t[2])" "6#-%\"AG6#&%\"tG6#\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "B(t[2])" "6#-%\"BG6#&%\"tG6#\"\"#" }{TEXT -1 31 " the \+ differential equation for " }{XPPEDIT 18 0 "x[1]" "6#&%\"xG6#\"\"\"" } {TEXT -1 30 " simplifies and can be solved." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 80 "simplify(ode[1], restrictions, convert([D[1](B)(t2) ,D[1](A)(t2), A(t2)], diff)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "combine(%, 'trig');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(-%%dif fG6$&%\"xG6#\"\"\"-%\"$G6$&%\"tGF*\"\"#F+F(F+*&#F+\"\"%F+*&-%$cosG6#,& *&\"\"$F+F/F+F+*&F;F+-%\"BG6#&F06#F1F+F+F+)-%\"AGF?F;F+F+F+\"\"!" }}} {PARA 0 "" 0 "" {TEXT -1 30 "Let us use the solutions that " } {XPPEDIT 18 0 "A(t[2])=A" "6#/-%\"AG6#&%\"tG6#\"\"#F%" }{TEXT -1 5 " a nd " }{XPPEDIT 18 0 "B(t[2])=alpha*t[2]" "6#/-%\"BG6#&%\"tG6#\"\"#*&%& alphaG\"\"\"&F(6#F*F-" }{TEXT -1 27 " and solve the equation of " } {XPPEDIT 18 0 "x[1]" "6#&%\"xG6#\"\"\"" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "subs(\{A(t2)=A,B(t2)=alpha*t2\}, %) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(-%%diffG6$&%\"xG6#\"\"\"-%\"$G 6$&%\"tGF*\"\"#F+F(F+*&#F+\"\"%F+*&-%$cosG6#,&*&\"\"$F+F/F+F+*(F;F+%&a lphaGF+&F06#F1F+F+F+)%\"AGF;F+F+F+\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "subs(xi[1](t1,t2)=Z(t1), %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(-%%diffG6$-%\"ZG6#&%\"tG6#\"\"\"-%\"$G6$F+\"\"#F.F(F .*&#F.\"\"%F.*&-%$cosG6#,&*&\"\"$F.F+F.F.*(F " 0 "" {MPLTEXT 1 0 38 "dsol ve(\{%, Z(0)=0, D(Z)(0)=0\}, Z(t1));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%\"ZG6#&%\"tG6#\"\"\",(*&#\"\"$\"#KF**(-%$sinGF&F*-F26#,$*(F.F*%&a lphaGF*&F(6#\"\"#F*F*F*)%\"AGF.F*F*F**&#F*F/F**(-%$cosGF&F*-FAF4F*F;F* F*!\"\"*&#F*F/F**&-FA6#,&*&F.F*F'F*F**(F.F*F7F*F8F*F*F*F;F*F*F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "combine(%, 'trig');" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"ZG6#&%\"tG6#\"\"\",(*&#F*\"#KF**& )%\"AG\"\"$F*-%$cosG6#,&F'!\"\"*(F2F*%&alphaGF*&F(6#\"\"#F*F*F*F*F**&# F*\"#;F**&F0F*-F46#,&F'F**(F2F*F9F*F:F*F*F*F*F7*&F-F**&-F46#,&*&F2F*F' F*F**(F2F*F9F*F:F*F*F*F0F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "eval(subs(t2=(omega*t-t1)/alpha,t1=t, %));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"ZG6#%\"tG,(*&#\"\"\"\"#KF+*&)%\"AG\"\"$F+-%$cosG 6#,&*&\"\"%F+F'F+!\"\"*(F0F+%&omegaGF+F'F+F+F+F+F+*&#F+\"#;F+*&F.F+-F2 6#,&*&\"\"#F+F'F+F7*(F0F+F9F+F'F+F+F+F+F7*&F*F+*&-F26#,$*(F0F+F9F+F'F+ F+F+F.F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "subs(A^3=8/ 3*A*alpha+4/3*F, %);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"ZG6#%\"tG ,(*&#\"\"\"\"#KF+*&,&**\"\")F+\"\"$!\"\"%\"AGF+%&alphaGF+F+*(\"\"%F+F1 F2%\"FGF+F+F+-%$cosG6#,&*&F6F+F'F+F2*(F1F+%&omegaGF+F'F+F+F+F+F+*&#F+ \"#;F+*&F.F+-F96#,&*&\"\"#F+F'F+F2*(F1F+F>F+F'F+F+F+F+F2*&F*F+*&-F96#, $*(F1F+F>F+F'F+F+F+F.F+F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 44 "We have o btained the following solution for " }{XPPEDIT 18 0 "x[1]" "6#&%\"xG6# \"\"\"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "rhs(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&#\"\"\"\"#KF&*&,&**\"\")F&\"\"$!\"\"%\"A GF&%&alphaGF&F&*(\"\"%F&F,F-%\"FGF&F&F&-%$cosG6#,&*&F1F&%\"tGF&F-*(F,F &%&omegaGF&F8F&F&F&F&F&*&#F&\"#;F&*&F)F&-F46#,&*&\"\"#F&F8F&F-*(F,F&F: F&F8F&F&F&F&F-*&F%F&*&-F46#,$*(F,F&F:F&F8F&F&F&F)F&F&F&" }}}{PARA 0 " " 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "A = (18*F+2*sqrt(-128*alpha ^3+81*F^2))^(1/3)/3+8/3*alpha/(18*F+2*sqrt(-128*alpha^3+81*F^2))^(1/3) " "6#/%\"AG,&*&),&*&\"#=\"\"\"%\"FGF+F+*&\"\"#F+-%%sqrtG6#,&*&\"$G\"F+ *$%&alphaG\"\"$F+!\"\"*&\"#\")F+*$F,F.F+F+F+F+*&F+F+F7F8F+F7F8F+**\"\" )F+F7F8F6F+),&*&F*F+F,F+F+*&F.F+-F06#,&*&F4F+*$F6F7F+F8*&F:F+*$F,F.F+F +F+F+*&F+F+F7F8F8F+" }{TEXT -1 12 " as before." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "So we have the following \+ approximation of our original problem" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "sol := A*cos(omega*t) + epsilon*%;" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#>%$solG,&*&%\"AG\"\"\"-%$cosG6#*&%&omegaGF(%\"tGF(F(F (*&%(epsilonGF(,(*&#F(\"#KF(*&,&**\"\")F(\"\"$!\"\"F'F(%&alphaGF(F(*( \"\"%F(F9F:%\"FGF(F(F(-F*6#,&*&F=F(F.F(F:*(F9F(F-F(F.F(F(F(F(F(*&#F(\" #;F(*&F6F(-F*6#,&*&\"\"#F(F.F(F:*(F9F(F-F(F.F(F(F(F(F:*&F3F(*&-F*6#,$* (F9F(F-F(F.F(F(F(F6F(F(F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 4 "with" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "assign(condA): 'A'=A;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG,&*&\"\"$!\"\",&*&\"#=\"\"\"%\"F GF,F,*&\"\"#F,,&*&\"$G\"F,)%&alphaGF'F,F(*&\"#\")F,)F-F/F,F,#F,F/F,#F, F'F,**\"\")F,F'F(F4F,F)#F(F'F," }}}{PARA 0 "" 0 "" {TEXT -1 5 "Take " }{XPPEDIT 18 0 "omega=1, epsilon=1" "6$/%&omegaG\"\"\"/%(epsilonGF%" } {TEXT -1 12 ", and hence " }{XPPEDIT 18 0 "alpha=0" "6#/%&alphaG\"\"! " }{TEXT -1 12 ". Take also " }{XPPEDIT 18 0 "F=1" "6#/%\"FG\"\"\"" } {TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "omega := 1: epsilon:=1: alpha:=0: F :=1: " }}}{PARA 0 "" 0 "" {TEXT -1 19 "Our so lution is now" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "simplify(so l);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&#\"\"\"\"\"$F&*&)\"\"'#\"\" #F'F&-%$cosG6#%\"tGF&F&F&*&#F&\"#CF&F-F&!\"\"*&#F&F3F&-F.6#,$*&F'F&F0F &F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&$\"+\\d(*e5!\"*\"\"\"-%$cosG6#%\"t GF(F(*&$\"+nmmmT!#6F(-F*6#,$*&$\"\"$\"\"!F(F,F(F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 43 "Th graph of this function looks as follows:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "plot(%, t=0..25);" }}{PARA 13 "" 1 "" {GLPLOT2D 339 288 288 {PLOTDATA 2 "6&-%'CURVESG6#7_v7$$\"\" !F)$\"35++n:Ck+6!#<7$$\"3\\LL$3FWYs#!#=$\"3ERt2+gT[5F,7$$\"3)pmm;a)G\\ aF0$\"3XL!p()yO$H!*F07$$\"3#***\\7`p)*>yF0$\"3Glu29P*=A(F07$$\"3SL$ek` o!>5F,$\"3p4#Q\")HPb8&F07$$\"3u;H#=XtB:\"F,$\"3e:xqG@t1RF07$$\"31+v=n$ ycG\"F,$\"3!4\\A;p'3kEF07$$\"3R$3_DG$)*=9F,$\"3eB_&=W@#=9F07$$\"3smm\" z>)G_:F,$\"3SYMF88iG7$$\"3'om\"zWnZ'o\"F,$!3'[Jt]z\"e!3\"F07$$\"3! omm;Hl1#=F,$!3&pc0S82ZL#F07$$\"3%pmT&QQ&[&>F,$!3qQo'e6Tre$F07$$\"35nmT &QU!*3#F,$!3apE1=/5H[F07$$\"3k$3_v:$fAAF,$!3/45?:`oPgF07$$\"3?+voHR9cB F,$!3-(yG%4#QI>(F07$$\"3u;H#=q%p*[#F,$!3K!>?(RArh#)F07$$\"3IL$eRZXKi#F ,$!3%)f!38?V_?*F07$$\"3q**\\(oz#)3(GF,$!3k?,&3,u!\\5F,7$$\"3bm;z>,_=JF ,$!3#G&>R&yg-5\"F,7$$\"3+LL$ekBE=$F,$!31FcmfeV*4\"F,7$$\"3%***\\(=A\\!pG L\\L#F07$$\"3=^7.K=B)4&F,$\"3^-z%H4iPg$F07$$\"3@ML$3_\"=M_F,$\"3M7)fke W;'[F07$$\"3U+]iS1ntaF,$\"330W88#)=!*pF07$$\"3^nmTg(fJr&F,$\"3)olq=:Nv &))F07$$\"3'RLekynF)fF,$\"3Nq,]OpSP5F,7$$\"3I,+]7eP_iF,$\"3]Z%3$**>'** 4\"F,7$$\"3/,]7yX0?jF,$\"3eqLRC\"o'*4\"F,7$$\"3o,+vVLt(Q'F,$\"3uk1;Wv# G4\"F,7$$\"3U,]P4@TbkF,$\"3a^OG&[O&z5F,7$$\"3=,++v34BlF,$\"3/O$3j9!)*f 5F,7$$\"3o++D1%[%emF,$\"3'y7K>$*=K+\"F,7$$\"3;++]Pf!Qz'F,$\"3=@D!4.h^D *F07$$\"3S++++![U#pF,$\"3r:5j4VYV$)F07$$\"3k++]i+paqF,$\"3`gkoKII4tF07 $$\"3'3++]7K^=(F,$\"3Y*=YVp1))='F07$$\"35,+](=ubJ(F,$\"3oGk+qz%Q,&F07$ $\"3W[_vF,$\"3())y1MW2r\"GF07$$ \"37](oa&e$4n(F,$\"3FGhdUS75@)F,$!3`!3ff)*oSM$F07$$\"3#\\LL3F-GN)F,$!3u>pp`._]YF07$$ \"3+NL3FWW\"f)F,$!3s@I+Q5(oy'F07$$\"3ILLL$e'3I))F,$!31L3Ac'HNo)F07$$\" 3(z;HK9%o2\"*F,$!3F`&[$)[u..\"F,7$$\"3)3+DJq\"G&Q*F,$!3))elr'RC&*4\"F, 7$$\"3'=/ED#>rY%*F,$!3dsv];vH+6F,7$$\"3i%3F>9U\"3&*F,$!3On@Ta&oc4\"F,7 $$\"3gD\"G8Os&p&*F,$!3e>t/q7p&3\"F,7$$\"3gm\"H2e-5j*F,$!3O,y_X2[q5F,7$ $\"3M]7`>I'Qv*F,$!3h:j/^q5D5F,7$$\"35MLLeMsw)*F,$!30SNe1bT:'*F07$$\"3> DJ?$3_6+\"!#;$!3@p?P%yW,u)F07$$\"3z;Hd?=j95F`al$!3!3=CbIOvr(F07$$\"3R3 F%zb6\"G5F`al$!3AJ`Ox#*G)e'F07$$\"3<+DJ&H\"fT5F`al$!3kWI%z(Hl*Q&F07$$ \"31](oH>FW0\"F`al$!3U[hIT%*o6UF07$$\"37+]i!4js1\"F`al$!3%oK43!*Ht,$F0 7$$\"3=]7G))*)4!3\"F`al$!3\")R#pj+tz\"=F07$$\"31+v$f)[$H4\"F`al$!33F1) **Q/c='FV7$$\"3U3xc)zFj5\"F`al$\"3$\\X(zy>pBjFV7$$\"3z;z>62s>6F`al$\"3 Y9j8gV&Q)=F07$$\"3:D\"GQi8J8\"F`al$\"3*e\\&G>^4NJF07$$\"3_L$ek`1l9\"F` al$\"3+Me4T4'*zVF07$$\"3]3x16a!)e6F`al$\"3#ek057Hd]&F07$$\"3[$3xcG/6< \"F`al$\"3!*\\Z=v,0(f'F07$$\"3YekGgJS$=\"F`al$\"3C#Q>&RPZIwF07$$\"3ULe *[.-d>\"F`al$\"3#H^DWc-yd)F07$$\"35](o/.MAA\"F`al$\"3cS\"R$\\*)H=5F,7$ $\"3ym;/Egw[7F`al$\"35LR9Fv?'4\"F,7$$\"3Ca8sZelb7F`al$\"3#fa)>(Qt05\"F ,7$$\"3qT5Spcai7F`al$\"3iH6uU:9)4\"F,7$$\"3;H23\"\\N%p7F`al$\"3QeD*faZ *)3\"F,7$$\"3!oTgFJDjF\"F`al$\"3S*)HXz]7t5F,7$$\"3)=z>h&\\5!H\"F`al$\" 37CT1<-fA5F,7$$\"3!o;z%*f%)QI\"F`al$\"3o9r7O_C$\\*F07$$\"3YLLeR](yK\"F `al$\"39w:1@i&3z(F07$$\"39+voza'=N\"F`al$\"3%3)3H)o,.u&F07$$\"3yTN'fk? [O\"F`al$\"3q'\\T)\\$z'fXF07$$\"3W$eRA\"exx8F`al$\"3_p$)Qd`]dLF07$$\"3 3Dc^y4t!R\"F`al$\"3quC3AxmZ@F07$$\"3tm;zWho.9F`al$\"35,nZ3QKp$*FV7$$\" 3-](=#*4qqT\"F`al$!3l8Z:>-/KJFV7$$\"3KLek`SXI9F`al$!3m`K#e;YOc\"F07$$ \"3y;H23!QQW\"F`al$!32d-369V9GF07$$\"33++]i>Ad9F`al$!3))y')*)fx3hSF07$ $\"3/]7y]bJq9F`al$!3EZ[)odgaE&F07$$\"3;+D1R\"4M[\"F`al$!3G<9xp.dNkF07$ $\"37]PMFF]'\\\"F`al$!38Bu!HJ%RWvF07$$\"32+]i:jf4:F`al$!3g)G2Vv6&e&)F0 7$$\"3:](oavL\\`\"F`al$!3=v!=O!QR65F,7$$\"31+DJ&>r-c\"F`al$!3e()*4@NAF 4\"F,7$$\"3g7.2AWIn:F`al$!3kG)y$Q%o(*4\"F,7$$\"3)\\7G)[wLu:F`al$!3;h() RMNu*4\"F,7$$\"3_Pfev3P\"e\"F`al$!3#3\">m>![E4\"F,7$$\"33]PM-TS)e\"F`a l$!3#)\\cpe'*ey5F,7$$\"3Z3%p\"F`al$!394:EXd-dJF07$$\"3\\L3x1*zvq\"F`al$!3q1l-JWE'*=F07$$ \"3@++v$4v5s\"F`al$!3+ynp8)y@N'FV7$$\"3!=HK*QPIL$RF07$$\"3!\\(=n8)eLy\"F`al$\"3 5!y-_>>Q;&F07$$\"3C$ekG&zs'z\"F`al$\"3goj.C%y>O'F07$$\"3&>Hd?4(45=F`al $\"3sD%[M\\`#)\\(F07$$\"3H++DJiYB=F`al$\"3MmC@^bQP&)F07$$\"3u;zWn[i[=F `al$\"3u=jWof745F,7$$\"3?Lek.Nyt=F`al$\"3OkJU))>s\"4\"F,7$$\"3w;aj4!e. )=F`al$\"3#4z/')oF\"*4\"F,7$$\"3'***\\i:D$p)=F`al$\"3da;zGAO+6F,7$$\"3 c\"z>'o(>-*=F`al$\"3GzTn[pl)4\"F,7$$\"3;$e9;-2N*=F`al$\"3E6k\"[;4a4\"F ,7$$\"3wu$4YF%z'*=F`al$\"39E`v&oH14\"F,7$$\"3rmTgF:3+>F`al$\"3G_SqdVL% 3\"F,7$$\"3ZLLeR0B8>F`al$\"3T!Q/\")**zW/\"F,7$$\"3B+Dc^&zj#>F`al$\"3`! >(\\xOOG)*F07$$\"3w;H#on!4_>F`al$\"3)o[1[8*)G6)F07$$\"3GLL3-=!y(>F`al$ \"3oNrWTNX`fF07$$\"3+Dcw%Qg7*>F`al$\"3Zu_9a-mKZF07$$\"3r;zWn*=Z+#F`al$ \"3g/Ec$>wc[$F07$$\"3V3-8]v<=?F`al$\"31_>r5r;HAF07$$\"3:+D\"G8O;.#F`al $\"3N%*QK'fXOr*FV7$$\"3&>/EvM)fW?F`al$!3a%\\UC)G\"RR#FV7$$\"3S$eRAcgv0 #F`al$!3=8f4%*GP]9F07$$\"3?DJ&pxA02#F`al$!3#ouvkv@=m#F07$$\"3+nmm\"*\\ [$3#F`al$!3GI]&[/o-(QF07$$\"3%oT&QG2u'4#F`al$!3e@f;PV)G4&F07$$\"3nmT5l k**4@F`al$!3i4eHwxr$G'F07$$\"3)o\"H#=?_K7#F`al$!3!fYWtv4eT(F07$$\"3sm; aQz]O@F`al$!37fsP9I>b%)F07$$\"3S]Pf$))*zi@F`al$!3n1o)Qo-#45F,7$$\"3sLe kG=4*=#F`al$!3EV%Qv2eM4\"F,7$$\"3i/wP?<8&>#F`al$!332k\">N0&*4\"F,7$$\" 3`v$4@hr6?#F`al$!3uRnoX\"R.5\"F,7$$\"33Y6%Q]6s?#F`al$!3o\\];!3]f4\"F,7 $$\"3*p\"Hd&R^K@#F`al$!3kB=\"GW(Q'3\"F,7$$\"3Xek.z6LDAF`al$!3?#4Gc%GA_ 5F,7$$\"3F++]i4TPAF`al$!3[uSIeqG$***F07$$\"3u$3_vvb7D#F`al$!3mrS^-)3`= *F07$$\"3'o;/Eb+^E#F`al$!3$Hj_w/O6?)F07$$\"3)*\\ilZ`%*yAF`al$!3Og0AD$4 ]3(F07$$\"3WL$3F9!z#H#F`al$!3kcv[_WWzeF07$$\"3i;zW#4t^I#F`al$!3'\\N!=] `KbZF07$$\"3;+v=Ugb**Q*HBF`al$!3gaR)>8 kKX#F07$$\"3)omm;%>KUBF`al$!3Q76&)Gv\"fH\"F07$$\"3MD\"G)[Z_bBF`al$!3)) )>DvAD_D'!#?7$$\"3y$e*)fbF(oBF`al$\"3#>,v#38vq6F07$$\"3CU5:j.$>Q#F`al$ \"3V " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 115 "It is in better agreement with the numerical approximation tha n the previos method when you consider the amplitude." }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 294 2 "9." }{TEXT -1 119 " Compute the determini ng system for Burgers' equation\n \+ " }{XPPEDIT 18 0 "u[t]=u[xx]+2*u*u[x]" "6# /&%\"uG6#%\"tG,&&F%6#%#xxG\"\"\"*(\"\"#F,F%F,&F%6#%\"xGF,F," }{TEXT -1 37 ",\nand complete the symmetry algebra.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(liesymm):" }}{PARA 7 "" 1 "" {TEXT -1 69 "Warning, the prot ected name close has been redefined and unprotected\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "Burgers_eqn := Diff(u(t,x), t) = Di ff(u(t,x),x$2) + 2*u(t,x)*Diff(u(t,x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,Burgers_eqnG/-%%DiffG6$-%\"uG6$%\"tG%\"xGF,,&-F'6$F) -%\"$G6$F-\"\"#\"\"\"*(F4F5F)F5-F'6$F)F-F5F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "eqns[1] := determine(Burgers_eqn, V, u(t,x), w); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>&%%eqnsG6#\"\"\"<+/-%%DiffG6$-%#V 1G6%%\"tG%\"xG%\"uGF1\"\"!/-F+6$F-F2F3/-F+6$F-F0,&*&\"\"#F'-F+6$-%#V2G F/F1F'F'-F+6$F--%\"$G6$F1F " 0 "" {MPLTEXT 1 0 29 "eqns[2] := autosimp(eqns[1]);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>&%%eqnsG6#\"\"#-%'&whereG6$<\"<./-%%V 3_2G6$%\"tG%\"xG,&*&F2\"\"\"%#C3GF5F5%#C4GF5/-%%V3_5G6#F1F7/-%%V3_4GF; F6/-%%V3_3GF;,&*(F'F5F1F5F6F5F5%#C6GF5/-%%V2_4GF;,&*(F'F5F1F5F7F5!\"\" %#C8GF5/-%%V2_3GF;,&*(F'F5F1F5F6F5FJ%$C10GF5/,&*&F'F5FDF5F5*&F'F5FQF5F 5\"\"!/-%#V2G6%F1F2%\"uG,(*&F2F5FOF5F5*(F'F5F1F5F7F5FJFKF5/-%%V3_1GF0F B/-%#V3GFZ,(*&FenF5FBF5F5F4F5F7F5/-%%V1_2GF;,(%$C11GF5*(F'F5)F1F'F5F6F 5FJ*(F'F5FQF5F1F5F5/-%#V1GFZFdo" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "eqns := op(2,eqns[2]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "sols := select(has, eqns, \{V1,V2,V3\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%solsG<%/-%#V2G6%%\"tG%\"xG%\"uG,(*&F+\"\"\",&*(\"\"# F/F*F/%#C3GF/!\"\"%$C10GF/F/F/*(F2F/F*F/%#C4GF/F4%#C8GF//-%#V3GF),(*&F ,F/,&*(F2F/F*F/F3F/F/%#C6GF/F/F/*&F+F/F3F/F/F7F//-%#V1GF),(%$C11GF/*(F 2F/)F*F2F/F3F/F4*(F2F/F5F/F*F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "sols := subs(C10=-C6, sols);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%solsG<%/-%#V3G6%%\"tG%\"xG%\"uG,(*&F,\"\"\",&*(\"\"# F/F*F/%#C3GF/F/%#C6GF/F/F/*&F+F/F3F/F/%#C4GF//-%#V2GF),(*&F+F/,&*(F2F/ F*F/F3F/!\"\"F4F>F/F/*(F2F/F*F/F6F/F>%#C8GF//-%#V1GF),(%$C11GF/*(F2F/) F*F2F/F3F/F>*(F2F/F4F/F*F/F>" }}}{PARA 0 "" 0 "" {TEXT -1 71 "We have \+ found a 5-dimensional symmetry algebra. Let us compute a basis:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "subs([C3=1, C4=0, C6=0, C8=0 , C11=0], sols);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/-%#V3G6%%\"tG% \"xG%\"uG,&*(\"\"#\"\"\"F*F.F(F.F.F)F./-%#V2GF',$*(F-F.F)F.F(F.!\"\"/- %#V1GF',$*&F-F.)F(F-F.F4" }}}{PARA 0 "" 0 "" {TEXT -1 26 "Solution cor responds with " }{XPPEDIT 18 0 "-2*t^2*diff(``,t)-2*x*t*diff(``,x)+(2* u*t+x)*diff(``,u)" "6#,(*(\"\"#\"\"\"*$%\"tGF%F&-%%diffG6$%!GF(F&!\"\" **F%F&%\"xGF&F(F&-F*6$F,F/F&F-*&,&*(F%F&%\"uGF&F(F&F&F/F&F&-F*6$F,F5F& F&" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "subs([C3=0, C4=1, C6=0 , C8=0, C11=0], sols);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/-%#V3G6%% \"tG%\"xG%\"uG\"\"\"/-%#V2GF',$*&\"\"#F+F(F+!\"\"/-%#V1GF'\"\"!" }}} {PARA 0 "" 0 "" {TEXT -1 26 "Solution corresponds with " }{XPPEDIT 18 0 "diff(``,u)-2*t*diff(``,x)" "6#,&-%%diffG6$%!G%\"uG\"\"\"*(\"\"#F)% \"tGF)-F%6$F'%\"xGF)!\"\"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "subs([C3=0, C4=0, C6=-1, C8=0, C11=0], sols);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/-%#V3G6%%\"tG%\"xG%\"uG,$F*!\"\"/-%#V2GF'F)/-%#V1GF' ,$*&\"\"#\"\"\"F(F6F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Solution corresponds with " }{XPPEDIT 18 0 "2*t*diff(``,t)+x*diff(``,x)-u*diff (``,u)" "6#,(*(\"\"#\"\"\"%\"tGF&-%%diffG6$%!GF'F&F&*&%\"xGF&-F)6$F+F- F&F&*&%\"uGF&-F)6$F+F1F&!\"\"" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "su bs([C3=0, C4=0, C6=0, C8=1, C11=0], sols);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/-%#V1G6%%\"tG%\"xG%\"uG\"\"!/-%#V3GF'F+/-%#V2GF'\"\" \"" }}}{PARA 0 "" 0 "" {TEXT -1 26 "Solution corresponds with " } {XPPEDIT 18 0 "diff(``,x)" "6#-%%diffG6$%!G%\"xG" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 44 "subs([C3=0, C4=0, C6=0, C8=0, C11=1], sols);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/-%#V2G6%%\"tG%\"xG%\"uG\"\"!/-%#V 3GF'F+/-%#V1GF'\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 26 "Solution corres ponds with " }{XPPEDIT 18 0 "diff(``,t" "6#-%%diffG6$%!G%\"tG" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 " " {TEXT 309 3 "10." }{TEXT -1 131 " Compute the determining system for the Boltzmann equation,\n \+ " }{XPPEDIT 18 0 "u[tx]+u[x]+u^2=0" "6#/,(& %\"uG6#%#txG\"\"\"&F&6#%\"xGF)*$F&\"\"#F)\"\"!" }{TEXT -1 81 ",\nand t ry to compute symmetry algebra. (Hint: you may have to load the proced ure " }{TEXT 0 11 "pdintegrate" }{TEXT -1 25 " of the \"hidden\" packa ge " }{TEXT 0 17 "liesymm/difftools" }{TEXT -1 2 ".)" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(liesymm):" }}{PARA 7 "" 1 "" {TEXT -1 69 "Warning, the protected name close has been redefi ned and unprotected\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "Bo ltzmann_eqn := Diff(u(t,x),t,x) + Diff(u(t,x),x) + u(t,x)^2 = 0;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%.Boltzmann_eqnG/,(-%%DiffG6%-%\"uG6$ %\"tG%\"xGF-F.\"\"\"-F(6$F*F.F/*$)F*\"\"#F/F/\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "eqns[1] := determine(Boltzmann_eqn, V, u(t, x), w);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>&%%eqnsG6#\"\"\"<./-%%Diff G6$-%#V1G6%%\"tG%\"xG%\"uGF2\"\"!/-F+6$F-F1F3/-F+6$-%#V2GF/F0F3/-F+6$F :F2F3/-F+6$F:-%\"$G6$F2\"\"#F3/-F+6%F-F2F1F3/-F+6$F-FBF3/-F+6$-%#V3GF/ F1,,*(FEF'FOF'F2F'!\"\"*&)F2FEF'-F+6$F:F1F'FS-F+6%FOF0F1FS*&-F+6$FOF2F 'FUF'F'*&FUF'-F+6$F-F0F'FS/-F+6%FOF0F2,&-F+6%F:F0F1F'FhnFS/-F+6$FOFB,& -F+6%F-F0F2F'-F+6%F:F2F1F'/-F+6%F:F0F2F3/-F+6%FOF2F1-F+6%F-F0F1" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "eqns[2] := autosimp(eqns[1], V, u);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>&%%eqnsG6#\"\"#-%'&whereG6 $<$/-%%DiffG6$-%%V1_2G6#%\"tGF3,&%#C1G!\"\"-%%V3_1GF2F6/-F.6$F7F3,&F5 \"\"\"F7F=<'/-%#V3G6%F3%\"xG%\"uG*&FDF=F7F=/-%%V3_2G6$F3FC\"\"!/-%#V1G FBF0/-%%V2_2G6#FC,&*&FCF=F5F=F=%#C2GF=/-%#V2GFBFR" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 26 "with(`liesymm/difftools`);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7=%'DsolveG%)autosimpG%&basicG%,constraintsG%(depvar sG%)describeG%(diffeqsG%*doubleintG%'dvalueG%+finddoubleG%-findfunctio nG%-findpartialsG%+findsimpleG%)fragmentG%+freezediffG%*indepvarsG%*ma kecanonG%)newcanonG%)nonbasicG%+nontrivialG%,pdintegrateG%(powsubsG%&r esetG%)separateG%*simpleintG%(trivialG%%vfixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "pdintegrate(eqns[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/-%#V3G6%%\"tG%\"xG%\"uG,&*&F*\"\"\"%#C1GF-!\"\"*(F*F --%$expG6#F(F-%$_C2GF-F-/-%#V1GF',&*&F1F-F4F-F/%$_C1GF-/-%#V2GF',&*&F) F-F.F-F-%#C2GF-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 88 "The symmetry algebra has 2 degrees of fre edom. The solution can be rewritten as follows:" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "V1(t,x,u)=_C1-_C2*exp(t), V2(t,x,u)=C1 *x+C2, V3(t,x,u)=_C2*u*exp(t)-C1*u" "6%/-%#V1G6%%\"tG%\"xG%\"uG,&%$_C1 G\"\"\"*&%$_C2GF,-%$expG6#F'F,!\"\"/-%#V2G6%F'F(F),&*&%#C1GF,F(F,F,%#C 2GF,/-%#V3G6%F'F(F),&*(F.F,F)F,-F06#F'F,F,*&F9F,F)F,F2" }}}}{MARK "0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }