{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 18 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 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 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 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 275 "" 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 "M aple 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 275 22 "(8) Estabilidad lineal" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 274 16 "Ejercicio 08-04 " }} {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 "c[2]" "6#&%\"cG6 #\"\"#" }{TEXT 264 8 " | " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"# " }{TEXT 265 2 " " }}{PARA 4 "" 0 "" {TEXT 266 6 " " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT 267 16 " | 0 " } {XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT 268 2 " " }}{PARA 4 "" 0 "" {TEXT -1 12 " ---------" }{TEXT 258 24 "----------------------- -" }}{PARA 4 "" 0 "" {TEXT 259 35 " | 0 0 1 " }}{PARA 4 "" 0 "" {TEXT -1 72 "a) Cu\341les deben ser los coeficient es para alcanzar al menos el orden 2 ?" }}{PARA 4 "" 0 "" {TEXT -1 35 "b) Es posible alcanzar el orden 3 ?" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 46 "(lo que antecede es Ejercicio del cap \355tulo 3)" }}{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" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "re start:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "interface(labelin g=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)=y 1,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 32 "k2_:=f(x0+c[2]*h,y0+c[2]*h*k1_):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "k3_:=f(x0+c[3]*h,y0+c[3]*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 263 2 "a)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 27 "condiciones para el orden 1" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 19 "Fi_0:=expand(k3_0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Fi_0G-%\"fG6$%#x0G%#y0G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "cond1a:=coeff(Y1,f(x0,y0))=coeff(Fi_0,f(x0,y0));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond1aG/\"\"\"F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 261 27 "condiciones para el orden 2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Fi10:=expand(k310);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Fi10G,&*&--&%\"DG6#\"\"\"6#%\"fG6$%#x0G%#y0GF,&%\"cG 6#\"\"$F,F,*(--&F*6#\"\"#F-F/F,F2F,-F.F/F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "cond2a:=(1/2)*coeff(Y2,D[1](f)(x0,y0))=coeff(Fi1 0,D[1](f)(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond2aG/#\"\" \"\"\"#&%\"cG6#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "con d2b:=(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#>%'cond2b G/#\"\"\"\"\"#&%\"cG6#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "sol1:=solve(\{cond2a,cond2b\},\{c[3]\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sol1G<#/&%\"cG6#\"\"$#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "c[3]:=subs(sol1,c[3]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>&%\"cG6#\"\"$#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 269 4 "b) " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 28 "condicion es para el orden 3 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Fi20 :=expand(k320);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Fi20G,,*&#\"\"\" \"\"%F(--&%\"DG6$F(F(6#%\"fG6$%#x0G%#y0GF(F(*&#F(\"\"#F(*&--&F-6$F(F6F /F1F(-F0F1F(F(F(*&F'F(*&--&F-6$F6F6F/F1F()F " 0 "" {MPLTEXT 1 0 76 "cond3a:=(1/6)*coeff(Y3,D[1,1](f)(x0,y0))=(1/2)*c oeff(Fi20,D[1,1](f)(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'con d3aG/#\"\"\"\"\"'#F'\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "cond3b:=(1/6)*coeff(coeff(Y3,D[1,2](f)(x0,y0)),f(x0,y0))=(1/2)*co eff(coeff(Fi20,D[1,2](f)(x0,y0)),f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3bG/#\"\"\"\"\"$#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(Y 3,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/#\" \"\"\"\"',$*&#F'\"\"#F'&%\"cG6#F,F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "cond3e:=(1/6)*coeff(coeff(Y3,D[2](f)(x0,y0)^2),f(x0, y0))=(1/2)*coeff(coeff(Fi20,D[2](f)(x0,y0)^2),f(x0,y0));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%'cond3eG/#\"\"\"\"\"',$*&#F'\"\"#F'&%\"cG6#F,F 'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 270 2 "c)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 271 23 "funci\363n de estabilidad " }}}{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 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#>%$k1_G*&%'lambdaG\"\"\"%#y0GF'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "k2_:=f(x0+c[2]*h,y0+c[2]*h*k 1_);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$k2_G*&%'lambdaG\"\"\",&%#y0 GF'**&%\"cG6#\"\"#F'%\"hGF'F&F'F)F'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "k3_:=f(x0+c[3]*h,y0+c[3]*h*k2_);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$k3_G*&%'lambdaG\"\"\",&%#y0GF'**&%\"cG6#\"\"$F'%\"hG F'F&F',&F)F'**&F,6#\"\"#F'F/F'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'&%\"cG6#\"\"$F'F&F'F'*,)F)F2F')F*F2F'F/F'&F06#F-F'F&F'F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "y1:=collect(subs(lambda=z/h, y1),y0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G*&,*\"\"\"F'%\"zGF'* &)F(\"\"#F'&%\"cG6#\"\"$F'F'*()F(F/F'F,F'&F-6#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-&%\"cG6#\"\"$F-F-*()F.F5 F-F2F-&F36#F1F-F-F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 272 51 "funci \363n de estabilidad para los metodos de orden 2 " }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 35 "r2:=unapply(subs(c[3]=1/2,r(z)),z);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r2Gf*6#%\"zG6\"6$%)operatorG%&arrow GF(,*\"\"\"F-9$F-*&#F-\"\"#F-*$)F.F1F-F-F-*&F0F-*&)F.\"\"$F-&%\"cG6#F1 F-F-F-F(F(F(" }}}}{MARK "0 0 0" 4 }{VIEWOPTS 1 1 0 3 4 1802 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }