{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 14 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 14 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 24 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 "" 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 "Heading 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 272 14 "Ejercicios de " }}{PARA 4 "" 0 "" {TEXT 274 22 "(8) Estabilidad lineal" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 273 16 "Ejercicio 08-03 " }} {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 18 " 1/2 | 1/2" }}{PARA 4 "" 0 "" {TEXT 260 17 " 1 | 1- " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT 268 5 " " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }}{PARA 4 "" 0 "" {TEXT -1 12 " --------- " }{TEXT 258 19 "-------------------" }}{PARA 4 "" 0 "" {TEXT 259 25 " | 1/6 " }{XPPEDIT 18 0 "b[2]" "6#&%\"bG6#\"\"#" } {TEXT 261 5 " " }{XPPEDIT 18 0 "b[3]" "6#&%\"bG6#\"\"$" }}{PARA 4 "" 0 "" {TEXT -1 38 "a) Para qu\351 valores de los par\341metros " } {XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "b [2]" "6#&%\"bG6#\"\"#" }{TEXT -1 4 " y " }{XPPEDIT 18 0 "b[3]" "6#&% \"bG6#\"\"$" }{TEXT -1 29 " los m\351todos son de orden 2 ?" }}{PARA 4 "" 0 "" {TEXT -1 37 "b) Existen valores de los par\341metros " } {XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "b [2]" "6#&%\"bG6#\"\"#" }{TEXT -1 4 " y " }{XPPEDIT 18 0 "b[3]" "6#&% \"bG6#\"\"$" }{TEXT -1 43 " para los que los m\351todos sean de orden \+ 3 ?" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 46 "(l o que antecede es Ejercicio del cap\355tulo 3)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 76 "c) Para la familia de m\351todos propuesta, calc\372lese la fun ci\363n de estabilidad" }}{PARA 4 "" 0 "" {TEXT -1 15 "en t\351rminos \+ de " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "b[2]" "6#&%\"bG6#\"\"#" }{TEXT -1 4 " y " }{XPPEDIT 18 0 "b[3] " "6#&%\"bG6#\"\"$" }{TEXT -1 60 " . Calc\372lese esta misma funci\363 n para los m\351todos de orden 2 " }}{PARA 4 "" 0 "" {TEXT -1 51 "y or den 3 que, en su caso, se hayan podido obtener." }}}{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:=subs(\{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 30 "k2_:=f(x0+1/2*h, y0+1/2*h*k1_):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "k3_:=f(x0 +h,y0+(1-alpha)*h*k1_+alpha*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 265 2 "a)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 27 "condiciones p ara el orden 1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Fi_0:=exp and(1/6*k1_0+b[2]*k2_0+b[3]*k3_0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%%Fi_0G,(*&#\"\"\"\"\"'F(-%\"fG6$%#x0G%#y0GF(F(*&&%\"bG6#\"\"#F(F*F(F (*&&F16#\"\"$F(F*F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "co nd1a:=coeff(Y1,f(x0,y0))=coeff(Fi_0,f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond1aG/\"\"\",(#F&\"\"'F&&%\"bG6#\"\"#F&&F+6#\"\"$F &" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 263 27 "condiciones para el orden 2 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Fi10:=expand(1/6*k110+b [2]*k210+b[3]*k310);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Fi10G,**&# \"\"\"\"\"#F(*&&%\"bG6#F)F(--&%\"DG6#F(6#%\"fG6$%#x0G%#y0GF(F(F(*&F'F( *(F+F(--&F1F-F3F5F(-F4F5F(F(F(*&&F,6#\"\"$F(F.F(F(*(F?F(F:F(F=F(F(" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "cond2a:=(1/2)*coeff(Y2,D[1] (f)(x0,y0))=coeff(Fi10,D[1](f)(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond2aG/#\"\"\"\"\"#,&*&F&F'&%\"bG6#F(F'F'&F,6#\"\"$F'" }}} {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'&% \"bG6#F(F'F'&F,6#\"\"$F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "sol1:=solve(\{cond1a,cond2a,cond2b\},\{alpha,b[2],b[3]\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sol1G<%/%&alphaGF'/&%\"bG6#\"\"##F,\"\"$/ &F*6#F.#\"\"\"\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "b[2] :=subs(sol1,b[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"bG6#\"\"##F '\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "b[3]:=subs(sol1,b [3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"bG6#\"\"$#\"\"\"\"\"'" } }}{EXCHG {PARA 0 "" 0 "" {TEXT 266 2 "b)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 28 "condiciones para el orden 3 " }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 43 "Fi20:=expand(1/6*k120+b[2]*k220+b[3]*k320);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Fi20G,,*&#\"\"\"\"\"$F(--&%\"DG6$F( F(6#%\"fG6$%#x0G%#y0GF(F(*&#\"\"#F)F(*&--&F-6$F(F6F/F1F(-F0F1F(F(F(*&F 'F(*&--&F-6$F6F6F/F1F()F " 0 "" {MPLTEXT 1 0 76 "cond3a:=(1/6)*coeff(Y3,D[1,1](f)(x0,y0))=(1/2 )*coeff(Fi20,D[1,1](f)(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%' cond3aG/#\"\"\"\"\"'F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 " cond3b:=(1/6)*coeff(coeff(Y3,D[1,2](f)(x0,y0)),f(x0,y0))=(1/2)*coeff(c oeff(Fi20,D[1,2](f)(x0,y0)),f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3bG/#\"\"\"\"\"$F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "cond3c:=(1/6)*coeff(coeff(Y3,D[2,2](f)(x0,y0)),f(x0,y0)^2)=(1 /2)*coeff(coeff(Fi20,D[2,2](f)(x0,y0)),f(x0,y0)^2);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%'cond3cG/#\"\"\"\"\"'F&" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 116 "cond3d:=(1/6)*coeff(coeff(Y3,D[2](f)(x0,y0)),D[1]( f)(x0,y0))=(1/2)*coeff(coeff(Fi20,D[2](f)(x0,y0)),D[1](f)(x0,y0));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3dG/#\"\"\"\"\"',$*&\"#7!\"\"%& alphaGF'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "cond3e:=(1/6 )*coeff(coeff(Y3,D[2](f)(x0,y0)^2),f(x0,y0))=(1/2)*coeff(coeff(Fi20,D[ 2](f)(x0,y0)^2),f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3 eG/#\"\"\"\"\"',$*&\"#7!\"\"%&alphaGF'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "sol2:=solve(\{cond3a,cond3b,cond3c,cond3d,cond3e\},\{ alpha\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sol2G<#/%&alphaG\"\"# " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "alpha:=subs(sol2,alpha) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 267 2 "c)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 269 23 "fun ci\363n de estabilidad " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "r estart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "interface(labeli ng=false):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f:=(x,y)->lam bda*y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"yG6\"6$%)o peratorG%&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#>%$k1_G*&%'lambdaG\"\"\"%#y0GF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "k2_:=f(x0+1/2*h,y0+1/2*h*k1_);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$k2_G*&%'lambdaG\"\"\",&%#y0GF'*&#F'\"\"#F'*(%\"hGF'F &F'F)F'F'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "k3_:=f(x0+ h,y0+(1-alpha)*h*k1_+alpha*h*k2_);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%$k3_G*&%'lambdaG\"\"\",(%#y0GF'**,&F'F'%&alphaG!\"\"F'%\"hGF'F&F'F)F 'F'**F,F'F.F'F&F',&F)F'*&#F'\"\"#F'*(F.F'F&F'F)F'F'F'F'F'F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "y1:=expand(y0+h*(1/6*k1_+b[2 ]*k2_+b[3]*k3_));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G,0%#y0G\"\" \"**\"\"'!\"\"%\"hGF'%'lambdaGF'F&F'F'**F+F'&%\"bG6#\"\"#F'F,F'F&F'F'* &#F'F1F'**F.F')F,F1F')F+F1F'F&F'F'F'**F+F'&F/6#\"\"$F'F,F'F&F'F'**F8F' F5F'F6F'F&F'F'*&F3F'*,F8F')F,F:F'%&alphaGF')F+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*&,0\"\"\"F'*&\"\"'!\"\"%\" zGF'F'*&&%\"bG6#\"\"#F'F+F'F'*&#F'F0F'*&F-F')F+F0F'F'F'*&&F.6#\"\"$F'F +F'F'*&F6F'F4F'F'*&F2F'*(F6F')F+F8F'%&alphaGF'F'F'F'%#y0GF'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "r:=unapply(collect(simplify( y1/y0),z),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rGf*6#%\"zG6\"6$% )operatorG%&arrowGF(,*\"\"\"F-*&#F-\"\"#F-*(&%\"bG6#\"\"$F-)9$F5F-%&al phaGF-F-F-*&,&*&F/F-&F36#F0F-F-F2F-F-)F7F0F-F-*&,(#F-\"\"'F-F " 0 "" {MPLTEXT 1 0 46 "r2:=unapply(subs(\{b[2]=2/3,b[3]=1/6\},r(z)),z);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r2Gf*6#%\"zG6\"6$%)operatorG%&arrow GF(,*\"\"\"F-*&#F-\"#7F-*&)9$\"\"$F-%&alphaGF-F-F-*&#F-\"\"#F-*$)F3F8F -F-F-F3F-F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 271 51 "funci\363n de estabilidad para los metodos de orden 3 " }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 37 "r3:=unapply(subs(\{alpha=2\},r2(z)),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r3Gf*6#%\"zG6\"6$%)operatorG%&arrowGF(,* \"\"\"F-*&#F-\"\"'F-*$)9$\"\"$F-F-F-*&#F-\"\"#F-*$)F3F7F-F-F-F3F-F(F(F (" }}}}{MARK "0 0 0" 5 }{VIEWOPTS 1 1 0 3 4 1802 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }