{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 "" 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 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 24 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 1 14 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 "" -1 268 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 1 14 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 "" -1 273 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 275 "" 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 "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 }} {SECT 0 {EXCHG {PARA 4 "" 0 "" {TEXT 261 14 "Ejercicios de " }}{PARA 4 "" 0 "" {TEXT 262 55 "(7) M\351todos de RUNGE-KUTTA impl\355citos y \+ semi-impl\355citos" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 260 51 "Ejercicio 07-13 (del EXAMEN ORDINARIO DE 24ENE11)" }} {PARA 4 "" 0 "" {TEXT -1 75 "Consid\351rese la familia de m\351todos i mpl\355citos de RUNGE-KUTTA con dos etapas " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 263 5 " " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT 264 11 " | " }{XPPEDIT 18 0 "alpha;" "6#%& alphaG" }{TEXT 265 13 " - 1 1 " }}{PARA 4 "" 0 "" {TEXT 266 20 " 1 | " }{XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT 273 13 " 0 " }}{PARA 4 "" 0 "" {TEXT -1 12 " ---------" } {TEXT 256 24 "------------------------" }}{PARA 4 "" 0 "" {TEXT 257 19 " | " }{XPPEDIT 18 0 "1/4;" "6#*&\"\"\"F$\"\"%!\"\" " }{TEXT 258 10 " " }{XPPEDIT 18 0 "3/4;" "6#*&\"\"$\"\"\"\" \"%!\"\"" }{TEXT 259 1 " " }}{PARA 4 "" 0 "" {TEXT -1 81 "a) B\372sque se el mayor orden alcanzable por los miembros de esta familia, indican do" }}{PARA 4 "" 0 "" {TEXT -1 16 "los valores de " }{XPPEDIT 18 0 "a lpha;" "6#%&alphaG" }{TEXT -1 8 " y de " }{XPPEDIT 18 0 "beta;" "6#% %betaG" }{TEXT -1 26 " para los que se consigue" }}{PARA 4 "" 0 "" {TEXT -1 40 "(desde aqu\355 es Ejercicio del cap\355tulo 8)" }}{PARA 4 "" 0 "" {TEXT -1 67 "b) B\372squese un m\351todo de la familia del e nunciado que sea A-estable" }}{PARA 4 "" 0 "" {TEXT -1 76 "c) Est\372d iese la A-estabilidad de los m\351todos de orden m\341ximo obtenidos a ntes" }}}{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 0 21 37 "# desarrollo de la verdadera \+ solucion" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "#y:=unapply(y(x ),x):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "#f:=unapply(f(x,y) ,x,y):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "f0:=f(x,y(x)):" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "f1:=subs(diff(y(x),x)=f(x, y(x)),diff(f0,x)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "f2:=s ubs(diff(y(x),x)=f(x,y(x)),diff(f1,x)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "f3:=subs(diff(y(x),x)=f(x,y(x)),diff(f2,x)):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "f0:=subs(\{y(x)=y0,x=x0\},f0 ):f0:=expand(f0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "f1:=su bs(\{y(x)=y0,x=x0\},f1):f1:=expand(f1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "f2:=subs(\{y(x)=y0,x=x0\},f2):f2:=expand(f2):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "f3:=subs(\{y(x)=y0,x=x0\},f3 ):f3:=expand(f3):" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 23 "# desar rollo del m\351todo" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "k10: =unapply(f(x0+alpha*h,y0+(alpha-1)*h*kp10(h)+h*kp20(h)),h):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "k20:=unapply(f(x0+h,y0+beta* h*kp10(h)),h):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "k100:=k10 (0):k100:=expand(k100):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 " k200:=k20(0):k200:=expand(k200):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Fi0:=expand((1/4)*k100+(3/4)*k200):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "k11:=D(k10):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 12 "k21:=D(k20):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "k110:=subs(\{kp10(0)=k100,kp20(0)=k200\},k11(0)):k110:=expand( k110):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "k210:=subs(\{kp10 (0)=k100,kp20(0)=k200\},k21(0)):k210:=expand(k210):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Fi1:=expand((1/4)*k110+(3/4)*k210):" }}} {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 98 "k120:=subs(\{D(kp10)(0)=k110,D(kp20)(0)=k210,k p10(0)=k100,kp20(0)=k200\},k12(0)):k120:=expand(k120):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "k220:=subs(\{D(kp10)(0)=k110,D(kp20 )(0)=k210,kp10(0)=k100,kp20(0)=k200\},k22(0)):k220:=expand(k220):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Fi2:=expand((1/4)*k120+(3/4) *k220):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "k13:=D(k12):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "k23:=D(k22):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "k130:=subs(\{(D@@2)(kp10)(0)=k120, (D@@2)(kp20)(0)=k220,D(kp10)(0)=k110,D(kp20)(0)=k210,kp10(0)=k100,kp20 (0)=k200\},k13(0)):k130:=expand(k130):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "k230:=subs(\{(D@@2)(kp10)(0)=k120,(D@@2)(kp20)(0)=k2 20,D(kp10)(0)=k110,D(kp20)(0)=k210,kp10(0)=k100,kp20(0)=k200\},k23(0)) :k230:=expand(k230):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Fi3 :=expand((1/4)*k130+(3/4)*k230):" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 24 "# consistencia y orden 1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "cond1a:=coeff(Fi0,f(x0,y0))=coeff(f0,f(x0,y0));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond1aG/\"\"\"F&" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 29 "# condiciones para el orden 2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "cond2a:=coeff(Fi1,D[1](f)(x0,y0))=c oeff((1/2)*f1,D[1](f)(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'c ond2aG/,&*&\"\"%!\"\"%&alphaG\"\"\"F+#\"\"$F(F+#F+\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "cond2b:=coeff(coeff(Fi1,D[2](f)(x0, y0)),f(x0,y0))=coeff(coeff((1/2)*f1,D[2](f)(x0,y0)),f(x0,y0));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond2bG/,&*&\"\"%!\"\"%&alphaG\"\" \"F+*(\"\"$F+F(F)%%betaGF+F+#F+\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "orden2:=solve(\{cond2a, cond2b\});" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%'orden2G<$/%&alphaG!\"\"/%%betaG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "alpha:=subs(orden2,alpha);beta:=sub s(orden2,beta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 29 "# condiciones para el orden 3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "cond3a:=coeff(Fi2,D[1,1](f)(x0,y0))=coeff ((1/3)*f2,D[1,1](f)(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'con d3aG/\"\"\"#F&\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "con d3b:=coeff(coeff(Fi2,D[1,2](f)(x0,y0)),f(x0,y0))=coeff(coeff((1/3)*f2, D[1,2](f)(x0,y0)),f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'con d3bG/\"\"##F&\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "cond 3c:=coeff(coeff(Fi2,D[2,2](f)(x0,y0)),f(x0,y0)^2)=coeff(coeff((1/3)*f2 ,D[2,2](f)(x0,y0)),f(x0,y0)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%' cond3cG/\"\"\"#F&\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 " cond3d:=coeff(coeff(Fi2,D[2](f)(x0,y0)),D[1](f)(x0,y0))=coeff(coeff((1 /3)*f2,D[2](f)(x0,y0)),D[1](f)(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3dG/\"\"!#\"\"\"\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "cond3e:=coeff(coeff(Fi2,D[2](f)(x0,y0)^2),f(x0,y0))= coeff(coeff((1/3)*f2,D[2](f)(x0,y0)^2),f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3eG/\"\"!#\"\"\"\"\"$" }}}{EXCHG {PARA 4 "" 0 " " {TEXT -1 73 "No se puede alcanzar orden 3 ; basta ver la primera de las condiciones, " }}{PARA 4 "" 0 "" {TEXT -1 46 "que es imposible. E l m\341ximo orden es 2 con " }{XPPEDIT 18 0 "alpha = -1;" "6#/%&alp haG,$\"\"\"!\"\"" }{TEXT -1 5 " y " }{XPPEDIT 18 0 "beta = 1;" "6#/% %betaG\"\"\"" }{TEXT -1 21 " " }}{PARA 4 "" 0 "" {TEXT -1 66 "Se obtiene orden m\341ximo 3 de una sola manera, para l os valores " }{XPPEDIT 18 0 "alpha = -1;" "6#/%&alphaG,$\"\"\"!\"\"" }{TEXT -1 5 " y " }{XPPEDIT 18 0 "beta = 1;" "6#/%%betaG\"\"\"" } {TEXT -1 9 " , o sea" }}{PARA 4 "" 0 "" {TEXT -1 14 "para el m\351tod o" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 270 34 " \+ -1 | -2 1" }}{PARA 4 "" 0 "" {TEXT 271 44 " \+ 1 | 1 0 " }}{PARA 4 "" 0 "" {TEXT -1 12 " ---------" }{TEXT 268 44 "------------------------------------- -------" }}{PARA 4 "" 0 "" {TEXT 269 21 " | " } {XPPEDIT 18 0 "1/4;" "6#*&\"\"\"F$\"\"%!\"\"" }{TEXT 274 10 " \+ " }{XPPEDIT 18 0 "3/4;" "6#*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT 275 1 " " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {MPLTEXT 0 21 30 "# funcion de esta bilidad para " }{XPPEDIT 18 0 "alpha = 1;" "6#/%&alphaG\"\"\"" } {MPLTEXT 0 21 3 " y " }{XPPEDIT 18 0 "beta = 1;" "6#/%%betaG\"\"\"" } {MPLTEXT 0 21 25 " (solo con orden 1 ) " }}}{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 17 "alpha:=1;beta:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "k10:=unapply(f(x0+alpha*h,y0+(alpha-1)*h* kp1(h)+h*kp2(h)),h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$k10Gf*6#%\" hG6\"6$%)operatorG%&arrowGF(*&%'lambdaG\"\"\",&%#y0GF.*&9$F.-%$kp2G6#F 2F.F.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "k20:=unapp ly(f(x0+beta*h,y0+beta*h*kp1(h)),h);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%$k20Gf*6#%\"hG6\"6$%)operatorG%&arrowGF(*&%'lambdaG\"\"\",&%#y0GF. *&9$F.-%$kp1G6#F2F.F.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "ka1:=unapply(f(x0+alpha*h,y0+(alpha-1)*h*kp1(h)+h*kp2(h)),h);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ka1Gf*6#%\"hG6\"6$%)operatorG%&arr owGF(*&%'lambdaG\"\"\",&%#y0GF.*&9$F.-%$kp2G6#F2F.F.F.F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "ka2:=unapply(f(x0+beta*h,y0+ beta*h*kp1(h)),h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ka2Gf*6#%\"hG 6\"6$%)operatorG%&arrowGF(*&%'lambdaG\"\"\",&%#y0GF.*&9$F.-%$kp1G6#F2F .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)=k 2\},ka2(h))\},\{k1,k2\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%testG< $/%#k2G,$*(%'lambdaG\"\"\"%#y0GF+,&*&F*F+%\"hGF+F+F+!\"\"F0F0/%#k1GF( " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "k1:=subs(test,k1);k2:=s ubs(test,k2);simplify(k1+3*k2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# k1G,$*(%'lambdaG\"\"\"%#y0GF(,&*&F'F(%\"hGF(F(F(!\"\"F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k2G,$*(%'lambdaG\"\"\"%#y0GF(,&*&F'F(%\"hGF( F(F(!\"\"F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\"%\"\"\"%'lamb daGF&%#y0GF&,&*&F'F&%\"hGF&F&F&!\"\"F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "y1:=expand(y0+h*((1/4)*k1+(3/4)*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:=c ollect(subs(lambda=z/h,y1),y0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# y1G*&,&\"\"\"F'*&%\"zGF',&F)F'F'!\"\"F+F+F'%#y0GF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "r:=unapply(collect(simplify(y1/y0),z),z):r_ particular:=r(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-r_particularG, $*&\"\"\"F',&%\"zGF'F'!\"\"F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 83 "# para estos valores alpha = 1 y beta = 1 el m\351todo es evidentemente A-estable " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {MPLTEXT 0 21 30 "# funcion de estabilidad para " }{XPPEDIT 18 0 "alpha = -1;" "6#/%&alphaG,$\"\"\"!\"\"" }{MPLTEXT 0 21 3 " y " } {XPPEDIT 18 0 "beta = 1;" "6#/%%betaG\"\"\"" }{MPLTEXT 0 21 23 " (orde n 2 m\341ximo) " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "rest art:" }}}{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$%\"xG%\"yG6\"6$%)oper atorG%&arrowGF)*&%'lambdaG\"\"\"9%F/F)F)F)" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 18 "alpha:=-1;beta:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG\"\"\"" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "k10: =unapply(f(x0+alpha*h,y0+(alpha-1)*h*kp1(h)+h*kp2(h)),h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$k10Gf*6#%\"hG6\"6$%)operatorG%&arrowGF(*&%'l ambdaG\"\"\",(%#y0GF.*(\"\"#F.9$F.-%$kp1G6#F3F.!\"\"*&F3F.-%$kp2GF6F.F .F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "k20:=unapply(f (x0+beta*h,y0+beta*h*kp1(h)),h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% $k20Gf*6#%\"hG6\"6$%)operatorG%&arrowGF(*&%'lambdaG\"\"\",&%#y0GF.*&9$ F.-%$kp1G6#F2F.F.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 " ka1:=unapply(f(x0+alpha*h,y0+(alpha-1)*h*kp1(h)+h*kp2(h)),h);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ka1Gf*6#%\"hG6\"6$%)operatorG%&arro wGF(*&%'lambdaG\"\"\",(%#y0GF.*(\"\"#F.9$F.-%$kp1G6#F3F.!\"\"*&F3F.-%$ kp2GF6F.F.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "ka2:= unapply(f(x0+beta*h,y0+beta*h*kp1(h)),h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ka2Gf*6#%\"hG6\"6$%)operatorG%&arrowGF(*&%'lambdaG\" \"\",&%#y0GF.*&9$F.-%$kp1G6#F2F.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<$/%#k1G,$**%'lambdaG\"\"\"%#y0GF+,&F+F+ *&F*F+%\"hGF+F+F+,(F+!\"\"*(\"\"#F+F*F+F/F+F1*&)F*F3F+)F/F3F+F+F1F1/%# k2G,$**F*F+F,F+,&F+F+*(\"\"$F+F*F+F/F+F+F+F0F1F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "k1:=subs(test,k1);k2:=subs(test,k2);simplify( k1+3*k2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k1G,$**%'lambdaG\"\"\" %#y0GF(,&F(F(*&F'F(%\"hGF(F(F(,(F(!\"\"*(\"\"#F(F'F(F,F(F.*&)F'F0F()F, F0F(F(F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k2G,$**%'lambdaG\"\" \"%#y0GF(,&F(F(*(\"\"$F(F'F(%\"hGF(F(F(,(F(!\"\"*(\"\"#F(F'F(F-F(F/*&) F'F1F()F-F1F(F(F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*,\"\"#\"\"\" %'lambdaGF&%#y0GF&,&F%F&*(\"\"&F&F'F&%\"hGF&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 37 "y1:=expand(y0+h*((1/4)*k1+(3/4)*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.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+*$)F)F-F'F'F+F+**\"\"&F'F-F+F)F-F*F+F+F' %#y0GF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "r:=unapply(colle ct(simplify(y1/y0),z),z):r_particular:=r(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-r_particularG,$*(\"\"#!\"\",(F'\"\"\"*&\"\"'F*%\"zGF *F**&\"\"$F*)F-F'F*F*F*,(F*F(*&F'F*F-F*F(*$F0F*F*F(F(" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 73 "# para estos valores alpha = -1 y \+ beta = 1 el m\351todo no es A-estable," }}{PARA 0 "" 0 "" {MPLTEXT 0 21 49 "ya que r(z) tiende a 3/2 cuando z tiende a " }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{MPLTEXT 0 21 3 " " }}}}{MARK "0 1 0" 7 }{VIEWOPTS 1 1 0 3 4 1802 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }