{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 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 }{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 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 56 "Ejercicio 03-04 (del EXAMEN EXTRAORDINARIO de 07JUL01)" }}{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#\"\"$" }{TEXT -1 43 " para los que los m\351todos sean de orden \+ 3 ?" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 40 "(d esde aqu\355 es Ejercicio del cap\355tulo 8)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 76 "c) Para la familia de m\351todo s propuesta, calc\372lese la funci\363n de estabilidad" }}{PARA 4 "" 0 "" {TEXT -1 15 "en t\351rminos de " }{XPPEDIT 18 0 "alpha" "6#%&alph aG" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "b[2]" "6#&%\"bG6#\"\"#" }{TEXT -1 4 " y " }{XPPEDIT 18 0 "b[3]" "6#&%\"bG6#\"\"$" }{TEXT -1 60 " . C alc\372lese esta misma funci\363n para los m\351todos de orden 2 " }} {PARA 4 "" 0 "" {TEXT -1 51 "y orden 3 que, en su caso, se hayan podid o 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,dif f(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 para el orden 1" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 43 "Fi_0:=expand(1/6*k1_0+b[2]*k2_0+b[3]*k3_0);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Fi_0G,(*&#\"\"\"\"\"'F(-%\"fG6$%#x0 G%#y0GF(F(*&&%\"bG6#\"\"#F(F*F(F(*&&F16#\"\"$F(F*F(F(" }}}{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 &\"\"'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#\"\"$#\"\"\"\"\"'" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 266 2 "b)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 28 "condic iones para el orden 3 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "F i20:=expand(1/6*k120+b[2]*k220+b[3]*k320);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Fi20G,,*&#\"\"\"\"\"$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)*coe ff(coeff(Y3,D[1,2](f)(x0,y0)),f(x0,y0))=(1/2)*coeff(coeff(Fi20,D[1,2]( f)(x0,y0)),f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3bG/# \"\"\"\"\"$F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "cond3c:=( 1/6)*coeff(coeff(Y3,D[2,2](f)(x0,y0)),f(x0,y0)^2)=(1/2)*coeff(coeff(Fi 20,D[2,2](f)(x0,y0)),f(x0,y0)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %'cond3cG/#\"\"\"\"\"'F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "cond3d:=(1/6)*coeff(coeff(Y3,D[2](f)(x0,y0)),D[1](f)(x0,y0))=(1/2 )*coeff(coeff(Fi20,D[2](f)(x0,y0)),D[1](f)(x0,y0));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%'cond3dG/#\"\"\"\"\"',$*&\"#7!\"\"%&alphaGF'F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "cond3e:=(1/6)*coeff(coeff(Y 3,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/#\"\"\"\"\"' ,$*&\"#7!\"\"%&alphaGF'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "sol2:=solve(\{cond3a,cond3b,cond3c,cond3d,cond3e\},\{alpha\});" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sol2G<#/%&alphaG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "alpha:=subs(sol2,alpha);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 267 2 "c)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 269 23 "funci\363n de estabilidad " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "interface(labeling=false): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f:=(x,y)->lambda*y;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"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 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*&%'la mbdaG\"\"\",(%#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#\"\"$F'F,F'F&F'F'**F8F'F5F'F6F'F&F'F'*&F 3F'*,F8F')F,F:F'%&alphaGF')F+F:F'F&F'F'F'" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 36 "y1:=collect(subs(lambda=z/h,y1),y0);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#y1G*&,0\"\"\"F'*&\"\"'!\"\"%\"zGF'F'*&&%\"bG6 #\"\"#F'F+F'F'*&#F'F0F'*&F-F')F+F0F'F'F'*&&F.6#\"\"$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$%)operatorG%&arrow GF(,*\"\"\"F-*&#F-\"\"#F-*(&%\"bG6#\"\"$F-)9$F5F-%&alphaGF-F-F-*&,&*&F /F-&F36#F0F-F-F2F-F-)F7F0F-F-*&,(#F-\"\"'F-F " 0 "" {MPLTEXT 1 0 46 "r2 :=unapply(subs(\{b[2]=2/3,b[3]=1/6\},r(z)),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r2Gf*6#%\"zG6\"6$%)operatorG%&arrowGF(,*\"\"\"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" 9 }{VIEWOPTS 1 1 0 3 4 1802 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }