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{SECT 0 {EXCHG {PARA 4 "" 0 "" {TEXT 268 14 "Ejercicios de " }}{PARA 
4 "" 0 "" {TEXT 287 22 "(8) Estabilidad lineal" }}{PARA 4 "" 0 "" 
{TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 286 18 "Ejercicio 08-21   " }
{TEXT 310 39 " (del EXAMEN EXTRAORDINARIO de 04JUL06)" }}{PARA 4 "" 0 
"" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 83 "a) B\372squense los m
\351todos 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 21 "     0  \+
     |      0" }}{PARA 4 "" 0 "" {TEXT 257 5 "     " }{XPPEDIT 18 0 "2
*alpha;" "6#*&\"\"#\"\"\"%&alphaGF%" }{TEXT 264 10 "   |      " }
{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT 272 9 "         " }
{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }}{PARA 4 "" 0 "" {TEXT -1 12 "   \+
---------" }{TEXT 258 24 "------------------------" }}{PARA 4 "" 0 "" 
{TEXT 259 20 "           |        " }{XPPEDIT 18 0 "beta;" "6#%%betaG
" }{TEXT 265 10 "          " }{XPPEDIT 18 0 "beta;" "6#%%betaG" }
{TEXT 266 1 " " }}{PARA 4 "" 0 "" {TEXT -1 8 " donde  " }{XPPEDIT 18 
0 "0 < alpha;" "6#2\"\"!%&alphaG" }{TEXT -1 6 "  y   " }{XPPEDIT 18 0 
"0 < beta;" "6#2\"\"!%%betaG" }{TEXT -1 21 "                     " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 46 "(lo que a
ntecede es  Ejercicio del cap\355tulo 7)" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 4 "" 0 "" {TEXT -1 80 "b) Calc\372lese la funci\363n de e
stabilidad para los m\351todos de la familia propuesta." }}{PARA 4 "" 
0 "" {TEXT -1 54 "Tambi\351n para los m\351todos del orden m\341ximo a
lcanzable  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 
-1 81 "c) Compru\351bese que dichos m\351todos de orden m\341ximo son \+
A-estables. B\372squese alg\372n " }}{PARA 4 "" 0 "" {TEXT -1 88 "otro
 m\351todo de la familia que tambi\351n sea A-estable, describiendo en
 este caso la regi\363n" }}{PARA 4 "" 0 "" {TEXT -1 14 "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,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:=sub
s(\{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 25 "k1_:=unapply(f(x0,y0),h):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 67 "k2_:=unapply(f(x0+2*alpha*h,y0+alpha*h*kp1_(h)+alph
a*h*kp2_(h)),h):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "k1_0:=k
1_(0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "k2_0:=k2_(0):" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "k11:=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(k
p1_)(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,$*(\"\"#\"\"\"%%betaGF(-%\"fG6$%#x0G%#y0GF(F(
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "cond1a:=coeff(Y1,f(x0,y
0))=coeff(Fi_0,f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond1a
G/\"\"\",$*&\"\"#F&%%betaGF&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 21 "solve(cond1a,\{beta\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#/
%%betaG#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 261 27 "condicion
es para el orden 2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Fi10:
=expand(beta*k110+beta*k210);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Fi
10G,&**\"\"#\"\"\"%%betaGF(--&%\"DG6#F(6#%\"fG6$%#x0G%#y0GF(%&alphaGF(
F(*,F'F(F)F(--&F-6#F'F/F1F(F4F(-F0F1F(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/#\"\"\"\"
\"#,$*(F(F'%%betaGF'%&alphaGF'F'" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 98 "cond2b:=(1/2)*coeff(coeff(Y2,D[2](f)(x0,y0)),f(x0,y0)
)=coeff(coeff(Fi10,D[2](f)(x0,y0)),f(x0,y0));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'cond2bG/#\"\"\"\"\"#,$*(F(F'%%betaGF'%&alphaGF'F'" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "solucion:=solve(\{cond1a,c
ond2a,cond2b\},\{alpha,beta\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)
solucionG<$/%%betaG#\"\"\"\"\"#/%&alphaGF(" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 262 28 "condiciones para el orden 3 " }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 34 "Fi20:=expand(beta*k120+beta*k220);" }}{PARA 12 "" 
1 "" {XPPMATH 20 "6#>%%Fi20G,,**\"\"%\"\"\"%%betaGF(--&%\"DG6$F(F(6#%
\"fG6$%#x0G%#y0GF()%&alphaG\"\"#F(F(*,\"\")F(F)F(--&F-6$F(F6F/F1F(F4F(
-F0F1F(F(*,F'F(F)F(F4F()F=F6F(--&F-6$F6F6F/F1F(F(*,F'F(F)F(--&F-6#F6F/
F1F(F4F(--&F-6#F(F/F1F(F(*,F'F(F)F()FEF6F(F4F(F=F(F(" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 76 "cond3a:=(1/6)*coeff(Y3,D[1,1](f)(x0,y0))=
(1/2)*coeff(Fi20,D[1,1](f)(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%'cond3aG/#\"\"\"\"\"',$*(\"\"#F'%%betaGF')%&alphaGF+F'F'" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "cond3b:=(1/6)*coeff(coeff(Y
3,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/#\"\"\"\"\"$
,$*(\"\"%F'%%betaGF')%&alphaG\"\"#F'F'" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 112 "cond3c:=(1/6)*coeff(coeff(Y3,D[2,2](f)(x0,y0)),f(x0,
y0)^2)=(1/2)*coeff(coeff(Fi20,D[2,2](f)(x0,y0)),f(x0,y0)^2);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%'cond3cG/#\"\"\"\"\"',$*(\"\"#F'%%betaGF')
%&alphaGF+F'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/#\"\"\"\"\"',$*(\"\"#F'%%betaGF')%&alphaGF+F'F'" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "cond3e:=(1/6)*coeff(coeff(Y
3,D[2](f)(x0,y0)^2),f(x0,y0))=(1/2)*coeff(coeff(Fi20,D[2](f)(x0,y0)^2)
,f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3eG/#\"\"\"\"\"'
,$*(\"\"#F'%%betaGF')%&alphaGF+F'F'" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 65 "solve(\{cond1a,cond2a,cond2b,cond3a,cond3b,cond3c,con
d3d,cond3e\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 269 95 "Es decir, no s
e alcanza orden  3 . El orden m\341ximo alcanzable es  2  y se alcanza
 con el m\351todo" }}{PARA 0 "" 0 "" {TEXT 273 19 "dado por el tablero
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 274 18 "    \+
 0    |      0" }}{PARA 4 "" 0 "" {TEXT 275 16 "     1    |     " }
{XPPEDIT 18 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 280 9 "         \+
" }{XPPEDIT 18 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }}{PARA 4 "" 0 "" 
{TEXT -1 12 "   ---------" }{TEXT 276 24 "------------------------" }}
{PARA 4 "" 0 "" {TEXT 277 17 "           |     " }{XPPEDIT 18 0 "1/2;
" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 278 9 "         " }{XPPEDIT 18 0 "1/
2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 279 1 " " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 270 2 "b)" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT 267 23 "funci\363n de estabilidad " }}}{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 25 "ka1:=
unapply(f(x0,y0),h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ka1Gf*6#%\"
hG6\"6$%)operatorG%&arrowGF(*&%'lambdaG\"\"\"%#y0GF.F(F(F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "ka2:=unapply(f(x0+2*alpha*h,y0+alph
a*h*kp1(h)+alpha*h*kp2(h)),h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$k
a2Gf*6#%\"hG6\"6$%)operatorG%&arrowGF(*&%'lambdaG\"\"\",(%#y0GF.*(%&al
phaGF.9$F.-%$kp1G6#F3F.F.*(F2F.F3F.-%$kp2GF6F.F.F.F(F(F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "test:=solve(\{k1=subs(\{kp1(h)=k1,k
p2(h)=k2\},ka1(h)),k2=subs(\{kp1(h)=k1,kp2(h)=k2\},ka2(h))\},\{k1,k2\}
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%testG<$/%#k1G*&%'lambdaG\"\"
\"%#y0GF*/%#k2G,$**F)F*F+F*,&F*F**(F)F*%&alphaGF*%\"hGF*F*F*,&F*!\"\"F
1F*F5F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "k1:=subs(test,k1
);k2:=subs(test,k2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k1G*&%'lamb
daG\"\"\"%#y0GF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k2G,$**%'lambda
G\"\"\"%#y0GF(,&F(F(*(F'F(%&alphaGF(%\"hGF(F(F(,&F(!\"\"F+F(F/F/" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "y1:=expand(y0+h*(beta*k1+bet
a*k2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G,*%#y0G\"\"\"**%\"hGF
'%%betaGF'%'lambdaGF'F&F'F'*,F)F'F*F'F+F'F&F',&F'!\"\"*(F+F'%&alphaGF'
F)F'F'F.F.*.F*F'F+\"\"#F&F'F-F.F0F'F)F2F." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 36 "y1:=collect(subs(lambda=z/h,y1),y0);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#y1G*&,*\"\"\"F'*&%%betaGF'%\"zGF'F'*(F)F'F*F'
,&F'!\"\"*&F*F'%&alphaGF'F'F-F-**F)F'F*\"\"#F,F-F/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$%
)operatorG%&arrowGF(*&,&\"\"\"!\"\"*&,&%&alphaGF.*&\"\"#F.%%betaGF.F/F
.9$F.F.F.,&F.F/*&F2F.F6F.F.F/F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
271 65 "que es la funci\363n de estabilidad de la familia general prop
uesta." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 294 8 "Cuando  " }{XPPEDIT 18 
0 "alpha;" "6#%&alphaG" }{TEXT 295 6 "   y  " }{XPPEDIT 18 0 "beta;" "
6#%%betaG" }{TEXT 296 69 "  valen  1/2 , o sea, en el \372nico caso de
 orden m\341ximo 2 , la funci\363n" }}{PARA 0 "" 0 "" {TEXT 297 19 "de
 estabilidad vale" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "r1:=un
apply(simplify(subs(solucion,r(z))),z);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#r1Gf*6#%\"zG6\"6$%)operatorG%&arrowGF(,$*&,&\"\"#\"\"\"9$F0F0
,&F/!\"\"F1F0F3F3F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 289 2 "c)" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT 288 18 "La A-estabilidad  " }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 290 8 "Cuando  " }{XPPEDIT 18 0 "alpha;" "6#%&al
phaG" }{TEXT 291 6 "   y  " }{XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT 
292 69 "  valen  1/2 , o sea, en el \372nico caso de orden m\341ximo 2
 , la funci\363n" }}{PARA 0 "" 0 "" {TEXT 293 20 "de estabil\361idad v
ale" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "r1:=unapply(simplify
(subs(solucion,r(z))),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r1Gf*6
#%\"zG6\"6$%)operatorG%&arrowGF(,$*&,&\"\"#\"\"\"9$F0F0,&F/!\"\"F1F0F3
F3F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 281 50 "el dominio de estabi
lidad, compuesto por los  z = " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" 
}{TEXT 282 35 "  h  tales que el m\363dulo de  r3(z) " }}{PARA 0 "" 0 
"" {TEXT 283 88 "sea inferior a  1 , estar\341 formado por los valores
 de  z  que disten menos del punto  -2" }}{PARA 0 "" 0 "" {TEXT 284 
86 "que del punto  2 . Esto corresponde al semiplano de parte real neg
ativa, por lo que el" }}{PARA 0 "" 0 "" {TEXT 285 30 "m\351todo estudi
ado es  A-estable" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 298 52 "Tomemos aho
ra cualquiera de los casos para los que  " }{XPPEDIT 18 0 "alpha = 2*b
eta;" "6#/%&alphaG*&\"\"#\"\"\"%%betaGF'" }{TEXT 299 25 " , lo que red
uce a  1  el" }}{PARA 0 "" 0 "" {TEXT 300 62 "numerador de la funci
\363n de estabilidad. Por ejemplo, hagamos  " }{XPPEDIT 18 0 "alpha = \+
1;" "6#/%&alphaG\"\"\"" }{TEXT 301 6 "  y   " }{XPPEDIT 18 0 "beta = 1
/2;" "6#/%%betaG*&\"\"\"F&\"\"#!\"\"" }{TEXT 302 13 "  . Para ese " }}
{PARA 0 "" 0 "" {TEXT 303 36 "caso, la funci\363n de estabilidad vale
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 304 14 "    \+
          " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "solucion2:=\{
alpha=1,beta=1/2\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*solucion2G<$
/%%betaG#\"\"\"\"\"#/%&alphaGF)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 46 "r2:=unapply(simplify(subs(solucion2,r(z))),z);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#r2Gf*6#%\"zG6\"6$%)operatorG%&arrowGF(,$*&\"\"
\"F.,&F.!\"\"9$F.F0F0F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 305 61 "A
hora, el dominio de estabilidad est\341 compuesto por los  z = " }
{XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT 306 22 "  h  cuya distanci
a al" }}{PARA 0 "" 0 "" {TEXT 307 84 "punto  1  sea superior a  1, que
 es el exterior de la circunferencia de centro  1  y" }}{PARA 0 "" 0 "
" {TEXT 308 85 "radio  1 (que pasa, pues, por el origen). Esta regi
\363n contiene al semiplano de parte " }}{PARA 0 "" 0 "" {TEXT 309 59 
"real negativa, por lo que el m\351todo estudiado es  A-estable" }}}}
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