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0 0 0 0 0 0 0 1 }{CSTYLE "" -1 305 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 306 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 307 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 308 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 309 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 310 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 311 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 312 "" 1 14 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 Outpu t" -1 12 1 {CSTYLE "" -1 -1 "Times" 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 267 14 "Ejercicios de " }}{PARA 4 "" 0 "" {TEXT 280 22 "(8) Estabilidad lineal" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 279 18 "Ejercicio 08-22 " } {TEXT 281 34 " (del EXAMEN ORDINARIO de 01FEB07)" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 83 "a) B\372squense los m\351 todos con el mayor orden posible para la familia de RUNGE-KUTTA " }} {PARA 4 "" 0 "" {TEXT -1 32 "impl\355citos del siguiente tablero" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 256 5 " " } {XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT 282 10 " | " } {XPPEDIT 18 0 "alpha" "6#%&alphaG" }}{PARA 4 "" 0 "" {TEXT 257 16 " \+ 1 | " }{XPPEDIT 18 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 270 9 " " }{XPPEDIT 18 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }} {PARA 4 "" 0 "" {TEXT -1 12 " ---------" }{TEXT 258 24 "------------ ------------" }}{PARA 4 "" 0 "" {TEXT 259 17 " | " } {XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT 264 10 " " } {XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT 265 1 " " }}{PARA 4 "" 0 "" {TEXT -1 8 " donde " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 6 " y " }{XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT -1 47 " son par \341metros arbitrarios " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 46 "(lo que antecede es Ejercicio del cap\355tulo 7)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 81 "b) Calc\372lese la funci\363n de estabilidad para los m \351todos de orden m\341ximo hallados." }}{PARA 4 "" 0 "" {TEXT -1 49 "\277Son A-estables dichos m\351todos de orden m\341ximo?. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 71 "c) B\372squese \+ alg\372n m\351todo de orden 1 de la familia que sea A-estable, " }} {PARA 4 "" 0 "" {TEXT -1 50 "describiendo en este caso la regi\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 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 49 "k1_:=unapply(f(x0+alpha*h,y0+alpha*h*kp1_(h)), h):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "k2_:=unapply(f(x0+h, y0+(1/2)*h*kp1_(h)+(1/2)*h*kp2_(h)),h):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "k1_0:=k1_(0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "k2_0:=k2_(0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "k1 1:=D(k1_):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "k21:=D(k2_): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "k110:=subs(\{kp1_(0)=k1 _0,kp2_(0)=k2_0\},k11(0)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "k210:=subs(\{kp1_(0)=k1_0,kp2_(0)=k2_0\},k21(0)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "k12:=D(k11):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 12 "k22:=D(k21):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "k120:=subs(\{D(kp1_)(0)=k110,D(kp2_)(0)=k210,kp1_(0)= k1_0,kp2_(0)=k2_0\},k12(0)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "k220:=subs(\{D(kp1_)(0)=k110,D(kp2_)(0)=k210,kp1_(0)=k1_0,kp2_(0 )=k2_0\},k22(0)):" }}}{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 34 "Fi_0:=expand(beta*k1_0+beta* k2_0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Fi_0G,$*(\"\"#\"\"\"%%bet aGF(-%\"fG6$%#x0G%#y0GF(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/\"\"\",$*&\"\"#F&%%betaGF&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "solve(cond1a,\{beta\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#/%%betaG#\"\"\"\"\"#" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 261 27 "condiciones para el orden 2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Fi10:=expand(beta*k110+beta*k210);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Fi10G,**(%%betaG\"\"\"--&%\"DG6#F(6#%\"fG 6$%#x0G%#y0GF(%&alphaGF(F(**F'F(--&F,6#\"\"#F.F0F(F3F(-F/F0F(F(*&F'F(F )F(F(*(F'F(F5F(F:F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "co nd2a:=(1/2)*coeff(Y2,D[1](f)(x0,y0))=coeff(Fi10,D[1](f)(x0,y0));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond2aG/#\"\"\"\"\"#,&*&%%betaGF'%& alphaGF'F'F+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'%&alphaGF'F'F+F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "solucion:=solve(\{cond1a,cond2a,cond2b\},\{alpha,beta \});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)solucionG<$/%%betaG#\"\"\" \"\"#/%&alphaG\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 28 "condicion es para el orden 3 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Fi20 :=expand(beta*k120+beta*k220);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%F i20G,:*(%%betaG\"\"\"--&%\"DG6$F(F(6#%\"fG6$%#x0G%#y0GF()%&alphaG\"\"# F(F(*,F5F(F'F(--&F,6$F(F5F.F0F(F3F(-F/F0F(F(**F'F(F3F()F;F5F(--&F,6$F5 F5F.F0F(F(*,F5F(F'F(--&F,6#F5F.F0F(F3F(--&F,6#F(F.F0F(F(*,F5F(F'F()FCF 5F(F3F(F;F(F(*&F'F(F)F(F(**F5F(F'F(F7F(F;F(F(*(F'F(F>F(F=F(F(**F'F(FCF (FGF(F4F(F(**F'F(FLF(F4F(F;F(F(*(F'F(FCF(FGF(F(*(F'F(FLF(F;F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "cond3a:=(1/6)*coeff(Y3,D[1,1 ](f)(x0,y0))=(1/2)*coeff(Fi20,D[1,1](f)(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3aG/#\"\"\"\"\"',&*&#F'\"\"#F'*&%%betaGF')%&alph aGF,F'F'F'*&F+F'F.F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 " cond3b:=(1/6)*coeff(coeff(Y3,D[1,2](f)(x0,y0)),f(x0,y0))=(1/2)*coeff(c oeff(Fi20,D[1,2](f)(x0,y0)),f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3bG/#\"\"\"\"\"$,&*&%%betaGF')%&alphaG\"\"#F'F'F+F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "cond3c:=(1/6)*coeff(coeff(Y 3,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'\"\"#F'*&%%betaGF')%&alphaGF,F'F'F'*&F+F'F.F'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/#\" \"\"\"\"',(*&%%betaGF')%&alphaG\"\"#F'F'*&#F'F.F'*&F+F'F-F'F'F'*&F0F'F +F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "cond3e:=(1/6)*coe ff(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/# \"\"\"\"\"',(*&%%betaGF')%&alphaG\"\"#F'F'*&#F'F.F'*&F+F'F-F'F'F'*&F0F 'F+F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "subs(solucion,[c ond1a,cond2a,cond2b,cond3a,cond3b,cond3c,cond3d,cond3e]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*/\"\"\"F%/#F%\"\"#F'F&/#F%\"\"'#F%\"\"%/#F%\" \"$F'F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "solve(\{cond 1a,cond2a,cond2b,cond3a,cond3b,cond3c,cond3d,cond3e\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 95 "Es decir, no se alcanza orden 3 . El or den m\341ximo alcanzable es 2 y se alcanza con el m\351todo" }} {PARA 0 "" 0 "" {TEXT 271 19 "dado por el tablero" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 272 18 " 0 | 0" }} {PARA 4 "" 0 "" {TEXT 273 16 " 1 | " }{XPPEDIT 18 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 278 9 " " }{XPPEDIT 18 0 "1/2; " "6#*&\"\"\"F$\"\"#!\"\"" }}{PARA 4 "" 0 "" {TEXT -1 12 " --------- " }{TEXT 274 24 "------------------------" }}{PARA 4 "" 0 "" {TEXT 275 17 " | " }{XPPEDIT 18 0 "1/2;" "6#*&\"\"\"F$\"\"#!\" \"" }{TEXT 276 9 " " }{XPPEDIT 18 0 "1/2;" "6#*&\"\"\"F$\"\"#! \"\"" }{TEXT 277 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 269 2 "b)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 266 57 "funci\363n de estabilidad para el m\351todo de orden 2 m\341ximo " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 284 8 "Cuando " }{XPPEDIT 18 0 "alp ha;" "6#%&alphaG" }{TEXT 285 13 " vale 0 y " }{XPPEDIT 18 0 "beta; " "6#%%betaG" }{TEXT 286 68 " vale 1/2 , o sea, en el \372nico caso \+ de orden m\341ximo 2 , la funci\363n" }}{PARA 0 "" 0 "" {TEXT 283 19 " de estabilidad vale" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f:=( x,y)->lambda*y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"y G6\"6$%)operatorG%&arrowGF)*&%'lambdaG\"\"\"9%F/F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "ka1:=unapply(subs(solucion,f(x0+alp ha*h,y0+alpha*h*kp1(h))),h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ka1 Gf*6#%\"hG6\"6$%)operatorG%&arrowGF(*&%'lambdaG\"\"\"%#y0GF.F(F(F(" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "ka2:=unapply(subs(solucion, f(x0+h,y0+(1/2)*h*kp1(h)+(1/2)*h*kp2(h))),h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ka2Gf*6#%\"hG6\"6$%)operatorG%&arrowGF(*&%'lambdaG\" \"\",(%#y0GF.*&#F.\"\"#F.*&9$F.-%$kp1G6#F5F.F.F.*&F2F.*&F5F.-%$kp2GF8F .F.F.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "test:=solv e(\{k1=subs(\{kp1(h)=k1,kp2(h)=k2\},ka1(h)),k2=subs(\{kp1(h)=k1,kp2(h) =k2\},ka2(h))\},\{k1,k2\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%test G<$/%#k1G*&%'lambdaG\"\"\"%#y0GF*/%#k2G,$**F)F*F+F*,&\"\"#F**&F)F*%\"h GF*F*F*,&F1!\"\"F2F*F5F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "k1:=subs(test,k1);k2:=subs(test,k2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k1G*&%'lambdaG\"\"\"%#y0GF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#k2G,$**%'lambdaG\"\"\"%#y0GF(,&\"\"#F(*&F'F(%\"hGF(F(F(,&F+!\"\"F,F( F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "y1:=expand(subs(sol ucion,y0+h*(beta*k1+beta*k2)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# y1G,*%#y0G\"\"\"**\"\"#!\"\"%\"hGF'%'lambdaGF'F&F'F'**F+F'F,F'F&F',&F) F**&F,F'F+F'F'F*F**,F)F*F,F)F&F'F.F*F+F)F*" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 36 "y1:=collect(subs(lambda=z/h,y1),y0);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#y1G*&,*\"\"\"F'*&\"\"#!\"\"%\"zGF'F'*&F+F',&F )F*F+F'F*F**(F)F*F+F)F-F*F*F'%#y0GF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "r:=unapply(collect(simplify(y1/y0),z),z):rz:=simplify (r(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#rzG,$*&,&\"\"#\"\"\"%\"z GF)F),&F(!\"\"F*F)F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT 287 50 "el dom inio de estabilidad, compuesto por los z = " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT 288 34 " h tales que el m\363dulo de r(z) " }} {PARA 0 "" 0 "" {TEXT 289 88 "sea inferior a 1 , estar\341 formado po r los valores de z que disten menos del punto -2" }}{PARA 0 "" 0 " " {TEXT 290 86 "que del punto 2 . Esto corresponde al semiplano de pa rte real negativa, por lo que el" }}{PARA 0 "" 0 "" {TEXT 291 30 "m \351todo estudiado es A-estable" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 293 2 "c)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 292 62 "funci\363n de estabilid ad para un m\351todo de orden exactamente 1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 294 53 "Tomemos ahora cualquiera de los casos para los que \+ " }{XPPEDIT 18 0 "beta = 1/2;" "6#/%%betaG*&\"\"\"F&\"\"#!\"\"" } {TEXT 296 26 " , de manera que el orden" }}{PARA 0 "" 0 "" {TEXT 297 48 "sea 1 al menos. Vamos a ensayar con el valor " }{XPPEDIT 18 0 " alpha = 1;" "6#/%&alphaG\"\"\"" }{TEXT 295 27 " , para el que no se ti ene " }}{PARA 0 "" 0 "" {TEXT 298 16 "orden 2 (m\341ximo)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 30 "solucion1:=\{alpha=1,beta=1/2\};" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%*solucion1G<$/%&alphaG\"\"\"/%%betaG#F(\"\"#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f:=(x,y)->lambda*y;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"yG6\"6$%)operatorG%& arrowGF)*&%'lambdaG\"\"\"9%F/F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "ka1:=unapply(subs(solucion1,f(x0+alpha*h,y0+alpha*h*k p1(h))),h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ka1Gf*6#%\"hG6\"6$%) operatorG%&arrowGF(*&%'lambdaG\"\"\",&%#y0GF.*&9$F.-%$kp1G6#F2F.F.F.F( F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "ka2:=unapply(subs(s olucion1,f(x0+h,y0+(1/2)*h*kp1(h)+(1/2)*h*kp2(h))),h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ka2Gf*6#%\"hG6\"6$%)operatorG%&arrowGF(*&%'lamb daG\"\"\",(%#y0GF.*&#F.\"\"#F.*&9$F.-%$kp1G6#F5F.F.F.*&F2F.*&F5F.-%$kp 2GF8F.F.F.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "test: =solve(\{k1=subs(\{kp1(h)=k1,kp2(h)=k2\},ka1(h)),k2=subs(\{kp1(h)=k1,k p2(h)=k2\},ka2(h))\},\{k1,k2\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% %testG<$/%#k1G,$*(%'lambdaG\"\"\"%#y0GF+,&F+!\"\"*&F*F+%\"hGF+F+F.F./% #k2GF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "k1:=subs(test,k1) ;k2:=subs(test,k2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k1G,$*(%'lam bdaG\"\"\"%#y0GF(,&F(!\"\"*&F'F(%\"hGF(F(F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k2G,$*(%'lambdaG\"\"\"%#y0GF(,&F(!\"\"*&F'F(%\"hGF(F (F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "y1:=expand(subs(so lucion1,y0+h*(beta*k1+beta*k2)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#y1G,&%#y0G\"\"\"**%\"hGF'%'lambdaGF'F&F',&F'!\"\"*&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*&,&\"\"\"F'*&%\"z GF',&F'!\"\"F)F'F+F+F'%#y0GF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "r:=unapply(collect(simplify(y1/y0),z),z):rz:=simplify(r(z));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#rzG,$*&\"\"\"F',&F'!\"\"%\"zGF'F)F) " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 299 61 "Ahora, el dominio de estabil idad est\341 compuesto por los z = " }{XPPEDIT 18 0 "lambda" "6#%'lam bdaG" }{TEXT 300 22 " h cuya distancia al" }}{PARA 0 "" 0 "" {TEXT 301 84 "punto 1 sea superior a 1, que es el exterior de la circunfe rencia de centro 1 y" }}{PARA 0 "" 0 "" {TEXT 302 85 "radio 1 (que \+ pasa, pues, por el origen). Esta regi\363n contiene al semiplano de pa rte " }}{PARA 0 "" 0 "" {TEXT 303 59 "real negativa, por lo que el m \351todo estudiado es A-estable" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 304 23 "Otra opcion: tomemos " }{XPPEDIT 18 0 "beta = 1/2;" "6#/%%betaG* &\"\"\"F&\"\"#!\"\"" }{TEXT 306 8 " , y " }{XPPEDIT 18 0 "alpha = 1 /2;" "6#/%&alphaG*&\"\"\"F&\"\"#!\"\"" }{TEXT 305 32 " , para el que t ampoco se tiene " }}{PARA 0 "" 0 "" {TEXT 307 16 "orden 2 (m\341ximo) " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "solucion1:=\{alpha=1/2,beta=1/2\}; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*solucion1G<$/%&alphaG#\"\"\"\" \"#/%%betaGF(" }}}{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 64 "ka1:=unapply(subs(solucion1,f(x0+alpha*h,y0+a lpha*h*kp1(h))),h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ka1Gf*6#%\"h G6\"6$%)operatorG%&arrowGF(*&%'lambdaG\"\"\",&%#y0GF.*&#F.\"\"#F.*&9$F .-%$kp1G6#F5F.F.F.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "ka2:=unapply(subs(solucion1,f(x0+h,y0+(1/2)*h*kp1(h)+(1/2)*h*kp2(h ))),h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ka2Gf*6#%\"hG6\"6$%)oper atorG%&arrowGF(*&%'lambdaG\"\"\",(%#y0GF.*&#F.\"\"#F.*&9$F.-%$kp1G6#F5 F.F.F.*&F2F.*&F5F.-%$kp2GF8F.F.F.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 99 "test:=solve(\{k1=subs(\{kp1(h)=k1,kp2(h)=k2\},ka1(h )),k2=subs(\{kp1(h)=k1,kp2(h)=k2\},ka2(h))\},\{k1,k2\});" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%%testG<$/%#k1G,$**\"\"#\"\"\"%'lambdaGF+%#y0GF +,&F*!\"\"*&F,F+%\"hGF+F+F/F//%#k2G,$**\"\"%F+F,F+F-F+,(F6F+*(F6F+F,F+ F1F+F/*&)F,F*F+)F1F*F+F+F/F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "k1:=subs(test,k1);k2:=subs(test,k2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k1G,$**\"\"#\"\"\"%'lambdaGF(%#y0GF(,&F'!\"\"*&F)F(% \"hGF(F(F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k2G,$**\"\"%\"\"\"% 'lambdaGF(%#y0GF(,(F'F(*(F'F(F)F(%\"hGF(!\"\"*&)F)\"\"#F()F-F1F(F(F.F( " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "y1:=expand(subs(solucio n1,y0+h*(beta*k1+beta*k2)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G ,(%#y0G\"\"\"**%\"hGF'%'lambdaGF'F&F',&\"\"#!\"\"*&F*F'F)F'F'F-F-*,F,F 'F)F'F*F'F&F',(\"\"%F'*(F1F'F*F'F)F'F-*&)F*F,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*&,(\"\"\"F'*&%\"zGF ',&\"\"#!\"\"F)F'F,F,*(F+F'F)F',(\"\"%F'*&F/F'F)F'F,*$)F)F+F'F'F,F'F'% #y0GF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "r:=unapply(collec t(simplify(y1/y0),z),z):rz:=factor(simplify(r(z)));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#rzG,$*&\"\"%\"\"\",&\"\"#!\"\"%\"zGF(!\"#F(" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 308 61 "Ahora, el dominio de estabilidad \+ est\341 compuesto por los z = " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG " }{TEXT 309 22 " h cuya distancia al" }}{PARA 0 "" 0 "" {TEXT 310 84 "punto 2 sea superior a 2, que es el exterior de la circunferenc ia de centro 2 y" }}{PARA 0 "" 0 "" {TEXT 311 93 "radio 2 (que pasa , pues, por el origen). Esta regi\363n tambi\351n contiene al semiplan o de parte " }}{PARA 0 "" 0 "" {TEXT 312 57 "real negativa, por lo que el m\351todo tambi\351n es A-estable" }}}}{MARK "0 3 0" 15 } {VIEWOPTS 1 1 0 3 4 1802 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }