{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 14 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 18 0 0 0 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 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 24 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 1 14 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 18 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 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Headi ng 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 4 4 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 4 "" 0 "" {TEXT 269 14 "Ejercicios de " }}{PARA 4 "" 0 "" {TEXT 274 22 "(8) Estabilidad lineal" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 268 51 "Ejercicio 08-07 (del E XAMEN ORDINARIO de 24ENE11)" }}{PARA 4 "" 0 "" {TEXT -1 81 "\nConsid \351rese la familia de m\351todos de RUNGE-KUTTA de 3 etapas y expl \355citos dados " }}{PARA 4 "" 0 "" {TEXT -1 14 "por el tablero" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 256 12 " 0 | " }}{PARA 4 "" 0 "" {TEXT 257 15 " 3/2 | " }{XPPEDIT 18 0 "beta" "6#%%betaG" }}{PARA 4 "" 0 "" {TEXT 260 16 " 0 | " } {XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT 270 6 " -1" }}{PARA 4 " " 0 "" {TEXT -1 12 " ---------" }{TEXT 258 19 "-------------------" }}{PARA 4 "" 0 "" {TEXT 259 33 " | 1/3 1/3 1/3" }} {PARA 4 "" 0 "" {TEXT -1 30 "a) D\355gase para qu\351 valores de " } {XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT -1 4 " y " }{XPPEDIT 18 0 " beta" "6#%%betaG" }{TEXT -1 28 " los m\351todos son de orden 2 " }} {PARA 4 "" 0 "" {TEXT -1 101 "b) De entre los anteriores m\351todos, b \372squense los que minimizan\nel coeficiente del t\351rmino del error ." }}{PARA 4 "" 0 "" {TEXT -1 46 "(lo que antecede es Ejercicio del c ap\355tulo 3)" }}{PARA 4 "" 0 "" {TEXT -1 64 "c) Calc\372lese la funci \363n de estabilidad de los m\351todos de orden 2" }}}{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 34 "k2_:=f(x0 +alpha*h,y0+alpha*h*k1_):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "k3_:=f(x0+h,y0+(1/2)*h*k1_+(1/2)*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 "" {MPLTEXT 1 0 17 "k12:=diff(k11,h):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "k22:=diff(k21,h):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "k32:=diff(k31,h):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "k120:=subs(h=0,k12):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "k220:=subs(h=0,k22):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "k320:=subs(h=0,k32):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 2 "a)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 261 27 "condiciones p ara el orden 1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Fi_0:=exp and(beta*k1_0+0*k2_0+(1-beta)*k3_0);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%%Fi_0G-%\"fG6$%#x0G%#y0G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "cond1a:=coeff(Y1,f(x0,y0))=coeff(Fi_0,f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond1aG/\"\"\"F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 27 "condiciones para el orden 2" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 45 "Fi10:=expand(beta*k110+0*k210+(1-beta)*k310);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Fi10G,*--&%\"DG6#\"\"\"6#%\"fG6$%#x 0G%#y0GF+*&--&F)6#\"\"#F,F.F+-F-F.F+F+*&%%betaGF+F&F+!\"\"*(F9F+F2F+F7 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'F'%%betaG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "cond2b:=(1/2)*coeff(coeff(Y2,D[2](f )(x0,y0)),f(x0,y0))=coeff(coeff(Fi10,D[2](f)(x0,y0)),f(x0,y0));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond2bG/#\"\"\"\"\"#,&F'F'%%betaG! \"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "sol2a:=solve(\{cond 2a,cond2b\},\{beta\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sol2aG<#/ %%betaG#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "beta :=subs(sol2a,beta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG#\"\"\" \"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 265 2 "b)" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 263 26 "coeficientes del termino " }{XPPEDIT 18 0 "1/3; " "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT 271 1 " " }{XPPEDIT 18 0 "f^2;" "6#* $%\"fG\"\"#" }{TEXT 272 2 "- " }{XPPEDIT 18 0 "Phi;" "6#%$PhiG" } {TEXT 273 25 "'(0) para el orden 3 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Fi20:=expand(beta*k120+0*k220+(1-beta)*k320);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Fi20G,,*&#\"\"\"\"\"#F(--&%\"DG6$F( F(6#%\"fG6$%#x0G%#y0GF(F(*&--&F-6$F(F)F/F1F(-F0F1F(F(*&F'F(*&--&F-6$F) F)F/F1F()F9F)F(F(F(*&F'F(*(--&F-6#F)F/F1F(--&F-6#F(F/F1F(%&alphaGF(F(F (*&F'F(*()FCF)F(FKF(F9F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "tau3a:=(1/3)*coeff(Y3,D[1,1](f)(x0,y0))-coeff(Fi20,D[1,1](f)(x0, y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&tau3aG#!\"\"\"\"'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "tau3b:=(1/3)*coeff(coeff(Y3 ,D[1,2](f)(x0,y0)),f(x0,y0))-coeff(coeff(Fi20,D[1,2](f)(x0,y0)),f(x0,y 0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&tau3bG#!\"\"\"\"$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "tau3c:=(1/3)*coeff(coeff(Y3 ,D[2,2](f)(x0,y0)),f(x0,y0)^2)-coeff(coeff(Fi20,D[2,2](f)(x0,y0)),f(x0 ,y0)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&tau3cG#!\"\"\"\"'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "tau3d:=(1/3)*coeff(coeff(Y3 ,D[2](f)(x0,y0)),D[1](f)(x0,y0))-coeff(coeff(Fi20,D[2](f)(x0,y0)),D[1] (f)(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&tau3dG,&#\"\"\"\"\" $F'*&\"\"#!\"\"%&alphaGF'F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "tau3e:=(1/3)*coeff(coeff(Y3,D[2](f)(x0,y0)^2),f(x0,y0))-coeff(coe ff(Fi20,D[2](f)(x0,y0)^2),f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%&tau3eG,&#\"\"\"\"\"$F'*&\"\"#!\"\"%&alphaGF'F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "sol3:=solve(\{tau3d,tau3e\},\{alpha\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sol3G<#/%&alphaG#\"\"#\"\"$" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 266 2 "c)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 267 36 "funci\363n de estabilidad para orden 2 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 26 "interface(labeling=false):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "beta:=1/2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f:=(x,y)->lambda*y;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"fGf*6$%\"xG%\"yG6\"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 34 "k2_:=f(x0+alpha*h,y0+alpha*h *k1_);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$k2_G*&%'lambdaG\"\"\",&%# y0GF'**%&alphaGF'%\"hGF'F&F'F)F'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "k3_:=f(x0+h,y0+(1/2)*h*k1_+(1/2)*h*k2_);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$k3_G*&%'lambdaG\"\"\",(%#y0GF'**\"\"#!\"\"% \"hGF'F&F'F)F'F'**F+F,F-F'F&F',&F)F'**%&alphaGF'F-F'F&F'F)F'F'F'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "y1:=expand(y0+h*(beta*k1_ +0*k2_+(1-beta)*k3_));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G,*%#y0 G\"\"\"*(%\"hGF'%'lambdaGF'F&F'F'**\"\"#!\"\"F)F,F*F,F&F'F'*,\"\"%F-F) \"\"$F*F0%&alphaGF'F&F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "y1:=collect(subs(lambda=z/h,y1),y0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G*&,*\"\"\"F'%\"zGF'*&#F'\"\"#F'*$)F(F+F'F'F'*&#F'\"\"%F'*& )F(\"\"$F'%&alphaGF'F'F'F'%#y0GF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "r:=unapply(collect(simplify(y1/y0),z),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rGf*6#%\"zG6\"6$%)operatorG%&arrowGF(,* \"\"\"F-9$F-*&#F-\"\"#F-*$)F.F1F-F-F-*&#F-\"\"%F-*&)F.\"\"$F-%&alphaGF -F-F-F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 1 0" 15 }{VIEWOPTS 1 1 0 3 4 1802 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }