{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 24 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 271 "" 1 14 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 18 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 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Ti mes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 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 270 48 "(3) Ecuaciones escalares: m\351todos de RUNGE- KUTTA" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 268 51 "Ejercicio 03-10 (del EXAMEN ORDINARIO de 24ENE11)" }}{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 16 " 0 | " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT 271 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 anterior es m\351todos, b\372squense los que minimizan\nel coeficiente del t \351rmino del error." }}{PARA 4 "" 0 "" {TEXT -1 40 "(desde aqu\355 es Ejercicio del cap\355tulo 8)" }}{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));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G-%\"fG6$%\"xG-%\"yG6#F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "y2:=subs(diff(y(x),x)=y1,diff(y1,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y2G,&--&%\"DG6#\"\"\"6#%\"fG6$%\"xG-%\"yG 6#F/F+*&--&F)6#\"\"#F,F.F+-F-F.F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "y3:=subs(diff(y(x),x)=y1,diff(y2,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y3G,*--&%\"DG6$\"\"\"F+6#%\"fG6$%\"xG-%\"yG6#F/ F+*&--&F)6$F+\"\"#F,F.F+-F-F.F+F+*&,&F4F+*&--&F)6$F8F8F,F.F+F9F+F+F+F9 F+F+*&--&F)6#F8F,F.F+,&--&F)6#F+F,F.F+*&FBF+F9F+F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Y1:=subs(\{y(x)=y0,x=x0\},y1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#Y1G-%\"fG6$%#x0G%#y0G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Y2:=subs(\{y(x)=y0,x=x0\},y2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#Y2G,&--&%\"DG6#\"\"\"6#%\"fG6$%#x0G %#y0GF+*&--&F)6#\"\"#F,F.F+-F-F.F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Y3:=subs(\{y(x)=y0,x=x0\},y3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#Y3G,*--&%\"DG6$\"\"\"F+6#%\"fG6$%#x0G%#y0GF+*&--&F)6 $F+\"\"#F,F.F+-F-F.F+F+*&,&F2F+*&--&F)6$F6F6F,F.F+F7F+F+F+F7F+F+*&--&F )6#F6F,F.F+,&--&F)6#F+F,F.F+*&F@F+F7F+F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "k1_:=f(x0,y0);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%$k1_G-%\"fG6$%#x0G%#y0G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "k2_:=f(x0+alpha*h,y0+alpha*h*k1_);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$k2_G-%\"fG6$,&%#x0G\"\"\"*&%&alphaGF*%\"hGF*F*,&%#y0GF**(F,F* F-F*-F&6$F)F/F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "k3_:=f (x0+h,y0+(1/2)*h*k1_+(1/2)*h*k2_);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%$k3_G-%\"fG6$,&%#x0G\"\"\"%\"hGF*,(%#y0GF**&#F*\"\"#F**&F+F*-F&6$F)F -F*F*F**&F/F**&F+F*-F&6$,&F)F**&%&alphaGF*F+F*F*,&F-F**(F:F*F+F*F2F*F* F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "k1_0:=subs(h=0,k1 _);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%k1_0G-%\"fG6$%#x0G%#y0G" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "k2_0:=subs(h=0,k2_);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%k2_0G-%\"fG6$%#x0G%#y0G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "k3_0:=subs(h=0,k3_);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%%k3_0G-%\"fG6$%#x0G%#y0G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "k11:=diff(k1_,h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$k11G\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "k21:=diff(k2_,h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$k21G,&*&-- &%\"DG6#\"\"\"6#%\"fG6$,&%#x0GF,*&%&alphaGF,%\"hGF,F,,&%#y0GF,*(F3F,F4 F,-F.6$F1F6F,F,F,F3F,F,*(--&F*6#\"\"#F-F/F,F3F,F8F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "k31:=diff(k3_,h);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$k31G,&--&%\"DG6#\"\"\"6#%\"fG6$,&%#x0GF+%\"hGF+,( %#y0GF+*&#F+\"\"#F+*&F1F+-F-6$F0F3F+F+F+*&F5F+*&F1F+-F-6$,&F0F+*&%&alp haGF+F1F+F+,&F3F+*(F@F+F1F+F8F+F+F+F+F+F+*&--&F)6#F6F,F.F+,(*&F5F+F8F+ F+*&F5F+F " 0 "" {MPLTEXT 1 0 20 "k110:=subs(h=0,k11); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%k110G\"\"!" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 20 "k210:=subs(h=0,k21);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%k210G,&*&--&%\"DG6#\"\"\"6#%\"fG6$%#x0G%#y0GF,%&alph aGF,F,*(--&F*6#\"\"#F-F/F,F2F,-F.F/F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "k310:=subs(h=0,k31);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%k310G,&--&%\"DG6#\"\"\"6#%\"fG6$%#x0G%#y0GF+*&--&F)6#\"\"#F,F.F+ -F-F.F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "k12:=diff(k11, h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$k12G\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "k22:=diff(k21,h);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$k22G,&*&,&*&--&%\"DG6$\"\"\"F.6#%\"fG6$,&%#x0GF.*&%& alphaGF.%\"hGF.F.,&%#y0GF.*(F5F.F6F.-F06$F3F8F.F.F.F5F.F.*(--&F,6$F.\" \"#F/F1F.F5F.F:F.F.F.F5F.F.*(,&*&F=F.F5F.F.*(--&F,6$FAFAF/F1F.F5F.F:F. F.F.F5F.F:F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "k32:=diff (k31,h);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$k32G,*--&%\"DG6$\"\"\"F +6#%\"fG6$,&%#x0GF+%\"hGF+,(%#y0GF+*&#F+\"\"#F+*&F1F+-F-6$F0F3F+F+F+*& F5F+*&F1F+-F-6$,&F0F+*&%&alphaGF+F1F+F+,&F3F+*(F@F+F1F+F8F+F+F+F+F+F+* &--&F)6$F+F6F,F.F+,(*&F5F+F8F+F+*&F5F+F " 0 "" {MPLTEXT 1 0 20 "k120:= subs(h=0,k12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%k120G\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "k220:=subs(h=0,k22);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%k220G,&*&,&*&--&%\"DG6$\"\"\"F.6#% \"fG6$%#x0G%#y0GF.%&alphaGF.F.*(--&F,6$F.\"\"#F/F1F.F4F.-F0F1F.F.F.F4F .F.*(,&*&F6F.F4F.F.*(--&F,6$F:F:F/F1F.F4F.F;F.F.F.F4F.F;F.F." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "k320:=subs(h=0,k32);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%k320G,*--&%\"DG6$\"\"\"F+6#%\"fG6$% #x0G%#y0GF+*&--&F)6$F+\"\"#F,F.F+-F-F.F+F+*&,&F2F+*&--&F)6$F6F6F,F.F+F 7F+F+F+F7F+F+*&--&F)6#F6F,F.F+,&*&--&F)6#F+F,F.F+%&alphaGF+F+*(F@F+FJF +F7F+F+F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 2 "a)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 261 27 "condiciones para el orden 1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Fi_0:=expand(beta*k1_0+0*k2_0+(1-be ta)*k3_0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Fi_0G-%\"fG6$%#x0G%#y 0G" }}}{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#>%'cond 1aG/\"\"\"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$%#x0G%#y0GF+*&--&F)6#\"\"#F,F.F+-F-F .F+F+*&%%betaGF+F&F+!\"\"*(F9F+F2F+F7F+F:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 66 "cond2a:=(1/2)*coeff(Y2,D[1](f)(x0,y0))=coeff(Fi10,D [1](f)(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond2aG/#\"\"\" \"\"#,&F'F'%%betaG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 " cond2b:=(1/2)*coeff(coeff(Y2,D[2](f)(x0,y0)),f(x0,y0))=coeff(coeff(Fi1 0,D[2](f)(x0,y0)),f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'con d2bG/#\"\"\"\"\"#,&F'F'%%betaG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "sol2a:=solve(\{cond2a,cond2b\},\{beta\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sol2aG<#/%%betaG#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "beta:=subs(sol2a,beta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 265 2 "b)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 263 26 "coeficie ntes del termino " }{XPPEDIT 18 0 "1/3;" "6#*&\"\"\"F$\"\"$!\"\"" } {TEXT 272 1 " " }{XPPEDIT 18 0 "f^2;" "6#*$%\"fG\"\"#" }{TEXT 273 2 "- " }{XPPEDIT 18 0 "Phi;" "6#%$PhiG" }{TEXT 274 25 "'(0) para el or den 3 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Fi20:=expand(beta *k120+0*k220+(1-beta)*k320);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Fi2 0G,,*&#\"\"\"\"\"#F(--&%\"DG6$F(F(6#%\"fG6$%#x0G%#y0GF(F(*&--&F-6$F(F) F/F1F(-F0F1F(F(*&F'F(*&--&F-6$F)F)F/F1F()F9F)F(F(F(*&F'F(*(--&F-6#F)F/ F1F(--&F-6#F(F/F1F(%&alphaGF(F(F(*&F'F(*()FCF)F(FKF(F9F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "tau3a:=(1/3)*coeff(Y3,D[1,1] (f)(x0,y0))-coeff(Fi20,D[1,1](f)(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&tau3aG#!\"\"\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "tau3b:=(1/3)*coeff(coeff(Y3,D[1,2](f)(x0,y0)),f(x0,y 0))-coeff(coeff(Fi20,D[1,2](f)(x0,y0)),f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&tau3bG#!\"\"\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "tau3c:=(1/3)*coeff(coeff(Y3,D[2,2](f)(x0,y0)),f(x0,y 0)^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,&#\"\"\"\"\"$F'*&\"\"#!\"\"%&alphaGF' F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "tau3e:=(1/3)*coeff(c oeff(Y3,D[2](f)(x0,y0)^2),f(x0,y0))-coeff(coeff(Fi20,D[2](f)(x0,y0)^2) ,f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&tau3eG,&#\"\"\"\"\"$ 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$%)operato rG%&arrowGF)*&%'lambdaG\"\"\"9%F/F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "k1_:=f(x0,y0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ 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*&%'lambda G\"\"\",(%#y0GF'**\"\"#!\"\"%\"hGF'F&F'F)F'F'**F+F,F-F'F&F',&F)F'**%&a lphaGF'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,*%#y0G\"\"\"*(%\"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(" }}}}{MARK "0 1 0" 13 }{VIEWOPTS 1 1 0 3 4 1802 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }