{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 1 236 0 76 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 48 37 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 "" -1 256 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 1 14 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 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 271 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 275 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 276 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 277 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 278 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 280 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 281 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 282 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 2 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 50 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Head ing 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 255 0 0 0 0 0 0 0 0 0 0 1 } 0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE " " -1 -1 "" 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 4 "" 0 "" {TEXT 273 14 "Ejercicios de " }}{PARA 4 "" 0 "" {TEXT 274 48 "(3) Ecuaciones escalares: m\351todos de RUNGE- KUTTA" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 272 51 "Ejercicio 03-08 (del EXAMEN ORDINARIO de 07FEB03)" }}{PARA 4 "" 0 " " {TEXT -1 86 "\nConsid\351rese el m\351todo gen\351rico expl\355cito \+ de Runge-Kutta dado por el siguiente tablero" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 256 13 " 0 | " }}{PARA 4 "" 0 "" {TEXT 257 6 " " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 263 8 " | " }{XPPEDIT 18 0 "a[21]" "6#&%\"aG6#\"#@" } {TEXT 264 2 " " }}{PARA 4 "" 0 "" {TEXT 265 6 " " }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 266 16 " | 0 " }{XPPEDIT 18 0 "a[32 ]" "6#&%\"aG6#\"#K" }{TEXT 267 2 " " }}{PARA 4 "" 0 "" {TEXT -1 12 " \+ ---------" }{TEXT 258 24 "------------------------" }}{PARA 4 "" 0 " " {TEXT 259 35 " | 0 0 1" }}{PARA 4 "" 0 " " {TEXT -1 92 "a) D\355gase cual es el orden m\341ximo que se puede al canzar y cu\341les deben ser los coeficientes " }}{PARA 4 "" 0 "" {TEXT -1 25 "para alcanzar dicho orden" }}{PARA 4 "" 0 "" {TEXT -1 79 "b) De entre los m\351todos de orden m\341ximo, d\355gase cu\341les so n los que minimizan el" }}{PARA 4 "" 0 "" {TEXT -1 19 "error de trunca ci\363n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 40 "(desde aqu\355 es Ejercicio del cap\355tulo 8)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 91 "c) Para la familia de m \351todos propuesta, calc\372lese la funci\363n de estabilidad. Calc \372lese la " }}{PARA 4 "" 0 "" {TEXT -1 80 "misma funci\363n para los m\351todos de orden m\341ximo y para los que minimizan el error" }}} {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 14 "y1:=f(x,y(x)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "y2:=subs(diff(y(x),x)=y1,diff(y1,x)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "y3:=subs(diff(y(x),x)=y1,diff(y2,x) ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Y1:=subs(\{y(x)=y0,x= x0\},y1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Y2:=subs(\{y(x )=y0,x=x0\},y2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Y3:=sub s(\{y(x)=y0,x=x0\},y3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " k1_:=f(x0,y0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "k2_:=f(x0 +(1/2)*h,y0+a[2,1]*h*k1_):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "k3_:=f(x0+h,y0+a[3,2]*h*k2_):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "k1_0:=subs(h=0,k1_):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "k2_0:=subs(h=0,k2_):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "k3_0:=subs(h=0,k3_):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "k11:=diff(k1_,h):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "k21:=diff(k2_,h):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "k31:=diff(k3_,h):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "k110:=subs(h=0,k11):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "k210:=subs(h=0,k21):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "k310:=subs(h=0,k31):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 2 "a)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 27 "condiciones p ara el orden 1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Fi_0:=exp and(k3_0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Fi_0G-%\"fG6$%#x0G%#y 0G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "cond1a:=coeff(Y1,f(x0 ,y0))=coeff(Fi_0,f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond 1aG/\"\"\"F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 275 27 "condiciones para el orden 2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Fi10:=expand (k310);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Fi10G,&--&%\"DG6#\"\"\"6 #%\"fG6$%#x0G%#y0GF+*(--&F)6#\"\"#F,F.F+&%\"aG6$\"\"$F6F+-F-F.F+F+" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "cond2a:=(1/2)*coeff(Y2,D[1] (f)(x0,y0))=coeff(Fi10,D[1](f)(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond2aG/#\"\"\"\"\"#F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "cond2b:=(1/2)*coeff(coeff(Y2,D[2](f)(x0,y0)),f(x0,y0))=coeff(c oeff(Fi10,D[2](f)(x0,y0)),f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%'cond2bG/#\"\"\"\"\"#&%\"aG6$\"\"$F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 95 "y no se puede conseguir el orden 2 de ninguna manera. El orden m\341ximo es 1 y cualquiera de los " }}{PARA 0 "" 0 "" {TEXT 277 39 "m\351todos de la familia tiene ese orden " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 276 2 "b)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 261 90 "Para \+ minimizar el error, de entre los m\351todos de orden 1 (que son todos) habr\341 que elegir " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "so l1:=solve(\{cond2b\},\{a[3,2]\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %%sol1G<#/&%\"aG6$\"\"$\"\"##\"\"\"F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "a[3,2]:=subs(sol1,a[3,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6$\"\"$\"\"##\"\"\"F(" }}}{EXCHG {PARA 4 "" 0 " " {TEXT 278 14 "y se obtiene " }{XPPEDIT 18 0 "a[32]" "6#&%\"aG6#\"#K " }{TEXT 279 3 " = " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" } {TEXT 280 57 " como valor para la minimizaci\363n del error, siendo a \372n " }{XPPEDIT 18 0 "a[21]" "6#&%\"aG6#\"#@" }{TEXT 281 12 " arbi trario" }{TEXT 282 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 269 2 "c)" }} }{EXCHG {PARA 0 "" 0 "" {TEXT 270 56 "funci\363n de estabilidad genera l y para el orden m\341ximo 1 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "interfac e(labeling=false):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f:=(x ,y)->lambda*y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"yG 6\"6$%)operatorG%&arrowGF)*&%'lambdaG\"\"\"9%F/F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "k1_:=f(x0,y0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$k1_G*&%'lambdaG\"\"\"%#y0GF'" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 35 "k2_:=f(x0+(1/2)*h,y0+a[2,1]*h*k1_);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$k2_G*&%'lambdaG\"\"\",&%#y0GF'**&%\"aG6$ \"\"#F'F'%\"hGF'F&F'F)F'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "k3_:=f(x0+h,y0+a[3,2]*h*k2_);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%$k3_G*&%'lambdaG\"\"\",&%#y0GF'**&%\"aG6$\"\"$\"\"#F'%\"hGF'F&F',&F )F'**&F,6$F/F'F'F0F'F&F'F)F'F'F'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "y1:=expand(y0+h*(k3_));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G,*%#y0G\"\"\"*(%\"hGF'%'lambdaGF'F&F'F'**)F)\"\"#F')F*F-F' &%\"aG6$\"\"$F-F'F&F'F'*,)F)F2F')F*F2F'F/F'&F06$F-F'F'F&F'F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "y1:=collect(subs(lambda=z/h, y1),y0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G*&,*\"\"\"F'%\"zGF'* &)F(\"\"#F'&%\"aG6$\"\"$F+F'F'*()F(F/F'F,F'&F-6$F+F'F'F'F'%#y0GF'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "r:=unapply(collect(simplify( y1/y0),z),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rGf*6#%\"zG6\"6$% )operatorG%&arrowGF(,*\"\"\"F-9$F-*&)F.\"\"#F-&%\"aG6$\"\"$F1F-F-*()F. F5F-F2F-&F36$F1F-F-F-F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 271 63 "f unci\363n de estabilidad para los metodos que minimizan el error " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "r2:=unapply(subs(a[3,2]=1/2, r(z)),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r2Gf*6#%\"zG6\"6$%)ope ratorG%&arrowGF(,*\"\"\"F-9$F-*&#F-\"\"#F-*$)F.F1F-F-F-*&F0F-*&)F.\"\" $F-&%\"aG6$F1F-F-F-F-F(F(F(" }}}}{MARK "0 0 0" 10 }{VIEWOPTS 1 1 0 3 4 1802 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }