{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 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 24 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 }{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 "Heading 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 }{PSTYLE "Maple Plot " -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 4 "" 0 "" {TEXT 271 14 "Ejercicios de " }}{PARA 4 "" 0 "" {TEXT 272 48 "(3) Ecuaciones escalares: m\351todos de RUNGE- KUTTA" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 270 51 "Ejercicio 03-03 (del EXAMEN ORDINARIO de 02FEB01)" }}{PARA 4 "" 0 " " {TEXT -1 81 "\nConsid\351rese la familia de m\351todos de Runge-Kutt a de 3 etapas y expl\355citos dados " }}{PARA 4 "" 0 "" {TEXT -1 14 "p or 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 23 " 0 | 1 -1" }}{PARA 4 "" 0 "" {TEXT -1 12 " ------- --" }{TEXT 258 16 "----------------" }}{PARA 4 "" 0 "" {TEXT 259 22 " \+ | 1/3 " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT 261 8 " 1/3" }}{PARA 4 "" 0 "" {TEXT -1 38 "a) Para qu\351 valores de \+ los par\341metros " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT -1 4 " \+ y " }{XPPEDIT 18 0 "beta" "6#%%betaG" }{TEXT -1 29 " los m\351todos s on de orden 2 ?" }}{PARA 4 "" 0 "" {TEXT -1 29 "b) Existe alguna elecc i\363n de " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT -1 4 " y " } {XPPEDIT 18 0 "beta" "6#%%betaG" }{TEXT -1 39 " que proporcione un m \351todo de orden 3 ?" }}{PARA 4 "" 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 76 "c) Para l a familia de m\351todos propuesta, calc\372lese la funci\363n de estab ilidad" }}{PARA 4 "" 0 "" {TEXT -1 15 "en t\351rminos de " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT -1 4 " y " }{XPPEDIT 18 0 "beta" "6# %%betaG" }{TEXT -1 2 " ." }}{PARA 4 "" 0 "" {TEXT -1 26 "d) En el supu esto de que " }{XPPEDIT 18 0 "abs(alpha)<2/3" "6#2-%$absG6#%&alphaG*& \"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 12 " y de que " }{XPPEDIT 18 0 "bet a=0" "6#/%%betaG\"\"!" }{TEXT -1 28 " , repres\351ntese la regi\363n d e" }}{PARA 4 "" 0 "" {TEXT -1 24 "estabilidad del m\351todo ." }}} {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 31 "k2_:=f(x0 +3/2*h,y0+beta*h*k1_):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "k 3_:=f(x0,y0+h*k1_-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(k 2_,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(1/3*k1_0+alpha*k2_0+1/3*k3_0);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Fi_0G,&*&#\"\"#\"\"$\"\"\"-%\"fG6$% #x0G%#y0GF*F**&%&alphaGF*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/\"\"\",&#\"\"#\"\"$F&%&alphaGF&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "sol1:=solve(cond1a,\{alpha\} );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sol1G<#/%&alphaG#\"\"\"\"\"$ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "alpha:=subs(sol1,alpha) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG#\"\"\"\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 263 27 "condiciones para el orden 2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Fi10:=expand(1/3*k110+alpha*k210+1/ 3*k310);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Fi10G,&*&#\"\"\"\"\"#F( --&%\"DG6#F(6#%\"fG6$%#x0G%#y0GF(F(*&#F(\"\"$F(*(--&F-6#F)F/F1F(%%beta GF(-F0F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "cond2a:=( 1/2)*coeff(Y2,D[1](f)(x0,y0))=coeff(Fi10,D[1](f)(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond2aG/#\"\"\"\"\"#F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "cond2b:=(1/2)*coeff(coeff(Y2,D[2](f)(x0,y0)), f(x0,y0))=coeff(coeff(Fi10,D[2](f)(x0,y0)),f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond2bG/#\"\"\"\"\"#,$*&\"\"$!\"\"%%betaGF'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "sol2:=solve(\{cond2a,cond 2b\},\{beta\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sol2G<#/%%betaG# \"\"$\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "beta:=subs(so l2,beta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG#\"\"$\"\"#" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 266 2 "b)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 28 "condiciones para el orden 3 " }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 43 "Fi20:=expand(1/3*k120+alpha*k220+1/3*k320);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Fi20G,,*&#\"\"$\"\"%\"\"\"--&%\"DG6 $F*F*6#%\"fG6$%#x0G%#y0GF*F**&#F(\"\"#F**&--&F.6$F*F7F0F2F*-F1F2F*F*F* *&F'F**&--&F.6$F7F7F0F2F*)F=F7F*F*F**&--&F.6#F7F0F2F*--&F.6#F*F0F2F*! \"\"*&)FFF7F*F=F*FN" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "cond 3a:=(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/#\"\"\"\"\"'#\"\"$\" \")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "cond3b:=(1/6)*coeff (coeff(Y3,D[1,2](f)(x0,y0)),f(x0,y0))=(1/2)*coeff(coeff(Fi20,D[1,2](f) (x0,y0)),f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3bG/#\" \"\"\"\"$#F(\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "cond3 c:=(1/6)*coeff(coeff(Y3,D[2,2](f)(x0,y0)),f(x0,y0)^2)=(1/2)*coeff(coef f(Fi20,D[2,2](f)(x0,y0)),f(x0,y0)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3cG/#\"\"\"\"\"'#\"\"$\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "cond3d:=(1/6)*coeff(coeff(Y3,D[2](f)(x0,y0)),D[1](f) (x0,y0))=(1/2)*coeff(coeff(Fi20,D[2](f)(x0,y0)),D[1](f)(x0,y0));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3dG/#\"\"\"\"\"'#!\"\"\"\"#" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "cond3e:=(1/6)*coeff(coeff( Y3,D[2](f)(x0,y0)^2),f(x0,y0))=(1/2)*coeff(coeff(Fi20,D[2](f)(x0,y0)^2 ),f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3eG/#\"\"\"\"\" '#!\"\"\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 2 "c)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 269 23 "funci\363n de estabilidad " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 26 "interface(labeling=false):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f:=(x,y)->lambda*y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF)*&%'lambd aG\"\"\"9%F/F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "k1_:= f(x0,y0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$k1_G*&%'lambdaG\"\"\"% #y0GF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "k2_:=f(x0+3/2*h,y 0+beta*h*k1_);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$k2_G*&%'lambdaG\" \"\",&%#y0GF'**%%betaGF'%\"hGF'F&F'F)F'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "k3_:=f(x0,y0+h*k1_-h*k2_);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$k3_G*&%'lambdaG\"\"\",(%#y0GF'*(%\"hGF'F&F'F)F'F'*(F +F'F&F',&F)F'**%%betaGF'F+F'F&F'F)F'F'F'!\"\"F'" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 45 "y1:=expand(y0+h*(1/3*k1_+alpha*k2_+1/3*k3_)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G,,%#y0G\"\"\"*,\"\"#F'\"\"$! \"\"%\"hGF'%'lambdaGF'F&F'F'**F,F'%&alphaGF'F-F'F&F'F'*,F/F')F-F)F'%%b etaGF')F,F)F'F&F'F'*,F*F+F,F*F-F*F2F'F&F'F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "y1:=collect(subs(lambda=z/h,y1),y0);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#y1G*&,,\"\"\"F'*&#\"\"#\"\"$F'%\"zGF'F'*&%&al phaGF'F,F'F'*(F.F')F,F*F'%%betaGF'F'*&#F'F+F'*&)F,F+F'F1F'F'!\"\"F'%#y 0GF'" }}}{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-*&)9$F0F-%%betaGF- F-!\"\"*(%&alphaGF-)F3\"\"#F-F4F-F-*&,&#F9F0F-F7F-F-F3F-F-F(F(F(" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 267 2 "d)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "beta:=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG\" \"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "r(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$*&,&#\"\"#\"\"$F$%&alphaGF$F$%\"zGF$F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "region:=abs(r(z))<1;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'regionG2-%$absG6#,&\"\"\"F**&,&#\" \"#\"\"$F*%&alphaGF*F*%\"zGF*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "div:=2/3+alpha;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% $divG,&#\"\"#\"\"$\"\"\"%&alphaGF)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "region:=abs(collect(r(z)/div,z))<1/div;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%'regionG2-%$absG6#,&%\"zG\"\"\"*&F+F+,&#\"\"# \"\"$F+%&alphaGF+!\"\"F+F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "radio:=1/div;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&radioG*&\"\"\" F&,&#\"\"#\"\"$F&%&alphaGF&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "with(plots):alpha:=1/3;\nimplicitplot((x+radio)^2+y^ 2=radio^2,x=-5..5,y=-3..3\n ,xtickmarks=[-1=`- ( 2 / 3 + a )`],ytickmarks=0,scaling=constrained,axesfont=[COURIER,BOLD,18]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG#\"\"\"\"\"$" }}{PARA 13 "" 1 "" {GLPLOT2D 836 580 580 {PLOTDATA 2 "6&-%'CURVESG6$X,%)anythingG6\" 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