{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 228 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 0 0 0 0 1 }{CSTYLE "" -1 256 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 260 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 1 14 25 1 1 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 265 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 1 18 0 0 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52 "# metodos de EULER y EULER implicito para un modelo " }}{PARA 0 "" 0 "" {MPLTEXT 0 21 53 "# lineal perturbado de ecuacion 'stiff', e l problema " }}{PARA 0 "" 0 "" {MPLTEXT 0 21 40 "# y' = - 50 ( y - co s(x) ) , y(0) = 0" }}{PARA 0 "" 0 "" {MPLTEXT 0 21 20 "# de soluci \363n exacta" }}{PARA 0 "" 0 "" {MPLTEXT 0 21 12 "# y(x) = " } {XPPEDIT 271 0 "2500/2501*cos(x)+50/2501*sin(x)-2500/2501*exp(-50*x); " "6#,(*(\"%+D\"\"\"\"%,D!\"\"-%$cosG6#%\"xGF&F&*(\"#]F&F'F(-%$sinG6#F ,F&F&*(F%F&F'F(-%$expG6#,$*&F.F&F,F&F(F&F(" }{MPLTEXT 0 21 3 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 0 21 46 "# que integramos en \+ [0,1.25] con diferentes " }}{PARA 0 "" 0 "" {MPLTEXT 0 21 21 "# valor es del paso h" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "interface(labeling=false) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Digits:=10:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 258 28 "EULER explicito (e. no aut.)" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "Euler:=proc(f:: procedure,ytotalini::list,nit::posint,h::numeric)\nlocal x0,y0,k,k1:\n x0:=ytotalini[1]:y0:=ytotalini[2]:\nfor k from 1 to nit do\nk1:=f(x0,y 0):\nx0:=x0+h:y0:=y0+h*k1:\nod:\nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 28 "EULER implicito (e. no aut.)" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 219 "EulerImplicito:=proc(f::pro cedure,ytotalini::list,nit::posint,h::numeric)\nlocal x0,y0,k,k1e,k1: \nx0:=ytotalini1[1]:y0:=ytotalini1[2]:\nfor k from 1 to nit do\nk1:=so lve(k1e=f(x0+h,y0+h*k1e)):\nx0:=x0+h:y0:=y0+h*k1:\nod:\nend:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 267 39 "EULER implicito aproximado (e. no aut.)" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 258 "EulerImplicitoAproximado:=proc(f::procedure,ytotalini::list,nit:: posint,h::numeric)\nlocal x0,y0,k,k1,k10,kt1:\nx0:=ytotalini[1]:y0:=yt otalini[2]:\nfor k from 1 to nit do\nkt1:=f(x0,y0):\nk10:=f(x0+h,y0+h* kt1):\nk1:=f(x0+h,y0+h*k10):\nx0:=x0+h:y0:=y0+h*k1:\nod:\nend:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 260 9 "Caso 1 : " }}{PARA 4 "" 0 "" {TEXT -1 99 "El problema escalar y' = - 50 ( y - cos(x) ), y(0) = 0 , del que se conoce la verdadera soluci\363n" }}{PARA 4 "" 0 "" {TEXT -1 7 " " }{XPPEDIT 18 0 "y(x) = 2500/2501*cos(x)+50/2501*s in(x)-2500/2501*exp(-50*x);" "6#/-%\"yG6#%\"xG,(*(\"%+D\"\"\"\"%,D!\" \"-%$cosG6#F'F+F+*(\"#]F+F,F--%$sinG6#F'F+F+*(F*F+F,F--%$expG6#,$*&F2F +F'F+F-F+F-" }}{PARA 4 "" 0 "" {TEXT 264 112 "se integra num\351ricame nte en el intervalo [ 0 , 1.25 ] con EULER , EULER implicito y EULER impl\355cito aproximado" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 265 44 "La e cuaci\363n (no aut.) y la condicion inicial" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "f:=unapply(-50.*(y-cos(x)),(x,y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),&*& $\"#]\"\"!\"\"\"9%F2!\"\"*&$F0F1F2-%$cosG6#9$F2F2F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 15 "Solucion exacta" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "ecu1:=diff(y(x),x)=f(x,y(x));ini:=y(0)=0;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ecu1G/-%%diffG6$-%\"yG6#%\"xGF,,&*& $\"#]\"\"!\"\"\"F)F2!\"\"*&$F0F1F2-%$cosGF+F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$iniG/-%\"yG6#\"\"!F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "solucion1:=dsolve(\{ecu1,ini\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*solucion1G/-%\"yG6#%\"xG,(*&#\"%+D\"%,D\"\"\"-%$cosG F(F/F/*&#\"#]F.F/-%$sinGF(F/F/*&#F-F.F/-%$expG6#,$*&F4F/F)F/!\"\"F/F> " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "u1:=unapply(subs(soluci on1,y(x)),x):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "gra1:=plot (u1,0..1.25):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 5 "Pasos" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "pasos1: =[30,40,70,30,30,40,70];h1:=[seq(1.25/pasos1[i],i=1..7)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'pasos1G7)\"#I\"#S\"#qF&F&F'F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#h1G7)$\"+nmmmT!#6$\"++++DJF($\"+'G9dy\"F(F&F& F)F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 261 44 "Caso 1a : EULER con paso 'grande' (30 pasos)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "yto talini1:=[0.,0.]:x1:=ytotalini1[1]:y1:=ytotalini1[2]:x11a[0]:=x1:y11a[ 0]:=y1: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "for i from 1 t o pasos1[1] do y1:=Euler(f,ytotalini1,1,h1[1]):x1:=x1+h1[1]:x11a[i]:=x 1:y11a[i]:=y1: ytotalini1:=[x1,y1]: od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "datos1a:=[seq([x11a[i],y11a[i]],i=0..pasos1[1])]:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "gra1a:=listplot(datos1a):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "display(gra1,gra1a,view=[ 0..1.25, -1..2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 396 396 396 {PLOTDATA 2 "6&-%'CURVESG6$7`o7$$\"\"!F)F(7$$\"3EL3FpE!Hq\"!#?$\"3YmIS]s4i\")!#> 7$$\"3_m;aQ`!eS$F-$\"3'[Cq>?(zl:!#=7$$\"3y*\\7y+3(3^F-$\"3c-XDRy>aAF67 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