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{SECT 0 {EXCHG {PARA 4 "" 0 "" {TEXT 263 14 "Ejercicios de " }}{PARA 
4 "" 0 "" {TEXT 267 22 "(8) Estabilidad lineal" }}{PARA 4 "" 0 "" 
{TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 266 56 "Ejercicio 08-14   (del E
XAMEN EXTRAORDINARIO de 01JUL05)" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}
{PARA 4 "" 0 "" {TEXT -1 59 "Est\372diese si el m\351todo impl\355cito
 de RUNGE-KUTTA del tablero" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
4 "" 0 "" {TEXT 256 5 "     " }{XPPEDIT 18 0 "1/4;" "6#*&\"\"\"F$\"\"%
!\"\"" }{TEXT 268 10 "    |     " }{XPPEDIT 18 0 "7/24;" "6#*&\"\"(\"
\"\"\"#C!\"\"" }{TEXT 269 6 "      " }{XPPEDIT 18 0 "-1/24;" "6#,$*&\"
\"\"F%\"#C!\"\"F'" }}{PARA 4 "" 0 "" {TEXT 257 5 "     " }{XPPEDIT 18 
0 "3/4;" "6#*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT 260 9 "   |     " }
{XPPEDIT 18 0 "13/24;" "6#*&\"#8\"\"\"\"#C!\"\"" }{TEXT 265 10 "      \+
    " }{XPPEDIT 18 0 "5/24;" "6#*&\"\"&\"\"\"\"#C!\"\"" }}{PARA 4 "" 
0 "" {TEXT -1 12 "   ---------" }{TEXT 258 24 "-----------------------
-" }}{PARA 4 "" 0 "" {TEXT 259 18 "           |      " }{XPPEDIT 18 0 
"1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 261 11 "           " }{XPPEDIT 
18 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 270 4 "    " }}{PARA 4 "" 
0 "" {TEXT -1 18 "es o no A-estable." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 
"interface(labeling=false):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 23 "f
unci\363n de estabilidad " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
19 "f:=(x,y)->lambda*y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%
\"xG%\"yG6\"6$%)operatorG%&arrowGF)*&%'lambdaG\"\"\"9%F/F)F)F)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "ka1:=unapply(f(x0+(1/4)*h,y0
+(7/24)*h*kp1(h)+(-1/24)*h*kp2(h)),h);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%$ka1Gf*6#%\"hG6\"6$%)operatorG%&arrowGF(*&%'lambdaG\"\"\",(%#y
0GF.*(#\"\"(\"#CF.9$F.-%$kp1G6#F5F.F.*&#F.F4F.*&F5F.-%$kp2GF8F.F.!\"\"
F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "ka2:=unapply(f(
x0+(3/4)*h,y0+(13/24)*h*kp1(h)+(5/24)*h*kp2(h)),h);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%$ka2Gf*6#%\"hG6\"6$%)operatorG%&arrowGF(*&%'lambdaG
\"\"\",(%#y0GF.*(#\"#8\"#CF.9$F.-%$kp1G6#F5F.F.*(#\"\"&F4F.F5F.-%$kp2G
F8F.F.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "test:=sol
ve(\{k1=subs(\{kp1(h)=k1,kp2(h)=k2\},ka1(h)),k2=subs(\{kp1(h)=k1,kp2(h
)=k2\},ka2(h))\},\{k1,k2\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%tes
tG<$/%#k1G,$**%'lambdaG\"\"\"%#y0GF+,&*&F*F+%\"hGF+F+\"\"%!\"\"F+,(*&)
F*\"\"#F+)F/F5F+F+\"#7F+*(\"\"'F+F*F+F/F+F1F1!\"$/%#k2G,$**F*F+F,F+,&F
0F+F.F+F+F2F1\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "k1:=s
ubs(test,k1);k2:=subs(test,k2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#
k1G,$**%'lambdaG\"\"\"%#y0GF(,&*&F'F(%\"hGF(F(\"\"%!\"\"F(,(*&)F'\"\"#
F()F,F2F(F(\"#7F(*(\"\"'F(F'F(F,F(F.F.!\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#k2G,$**%'lambdaG\"\"\"%#y0GF(,&\"\"%F(*&F'F(%\"hGF(F
(F(,(*&)F'\"\"#F()F-F1F(F(\"#7F(*(\"\"'F(F'F(F-F(!\"\"F6\"\"$" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "y1:=expand(y0+h*((1/2)*k1+(1
/2)*k2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G,&%#y0G\"\"\"*,\"#7
F'%\"hGF'%'lambdaGF'F&F',(*&)F+\"\"#F')F*F/F'F'F)F'*(\"\"'F'F+F'F*F'!
\"\"F3F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "y1:=collect(sub
s(lambda=z/h,y1),y0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G*&,&\"
\"\"F'*(\"#7F'%\"zGF',(*$)F*\"\"#F'F'F)F'*&\"\"'F'F*F'!\"\"F1F'F'%#y0G
F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "r:=unapply(collect(si
mplify(y1/y0),z),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rGf*6#%\"z
G6\"6$%)operatorG%&arrowGF(*&,(*$)9$\"\"#\"\"\"F2\"#7F2*&\"\"'F2F0F2F2
F2,(F.F2F3F2*&F5F2F0F2!\"\"F8F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
264 39 "es la funci\363n de estabilidad del m\351todo" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 271 25 "A-estabilidad del m\351todo " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "solve(z^2+6*z+12);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6$,&!\"$\"\"\"*&-%%sqrtG6#\"\"$F%^#F%F%F%,&F$F%*&^#!
\"\"F%F'F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "solve(z^2-6
*z+12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&\"\"$\"\"\"*&-%%sqrtG6#F$
F%^#F%F%F%,&F$F%*&^#!\"\"F%F'F%F%" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 82 "r:=unapply(((z+3-I*sqrt(3))*(z+3+I*sqrt(3)))/((z-3-I*
sqrt(3))*(z-3+I*sqrt(3))),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"r
Gf*6#%\"zG6\"6$%)operatorG%&arrowGF(**,(9$\"\"\"\"\"$F/*&^#!\"\"F/-%%s
qrtG6#F0F/F/F/,(F.F/F0F/*&F4F/^#F/F/F/F/,(F.F/F0F3F1F/F3,(F.F/F0F3F8F/
F3F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "r:=unapply(((z+
3)^2+3)/((z-3)^2+3),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rGf*6#%
\"zG6\"6$%)operatorG%&arrowGF(*&,&*$),&9$\"\"\"\"\"$F2\"\"#F2F2F3F2F2,
&*$),&F1F2F3!\"\"F4F2F2F3F2F9F(F(F(" }}}{EXCHG {PARA 4 "" 0 "" {TEXT 
-1 101 "Para  z  real, (z + 3)^2  es el cuadrado de la distancia de z \+
a  -3 y lo an\341logo con (z - 3)^2  y   3" }}{PARA 4 "" 0 "" {TEXT 
-1 91 "Los n\372meros reales negativos est\341n en el dominio  D  y lo
s positivos en su complementario. " }}{PARA 4 "" 0 "" {TEXT -1 84 "Per
o no hay en el caso complejo la misma interpretaci\363n sobre la dista
ncia mientras " }}{PARA 4 "" 0 "" {TEXT -1 93 "no intervenga el m\363d
ulo, as\355 que todav\355a no podemos concluir si el m\351todo es o no
 A-estable. " }}{PARA 4 "" 0 "" {TEXT -1 96 "Vamos a construir la fron
tera de  D  de ecuaci\363n   | r(z) |^2 = 1 en el plano complejo, o lo
 que" }}{PARA 4 "" 0 "" {TEXT -1 11 "es lo mismo" }}{PARA 4 "" 0 "" 
{TEXT -1 43 "      |(z + 3)^2 + 3|^2 = |(z - 3)^2 + 3|^2" }}{PARA 4 "
" 0 "" {TEXT 272 42 "poniendo  z=x+i y , lo anterior se escribe" }}
{PARA 4 "" 0 "" {TEXT 273 92 "      ((x + 3)^2 - y^2 + 3)^2 + 4 y^2 (x
 + 3)^2 = ((x - 3)^2 - y^2 + 3)^2 + 4 y^2 (x - 3)^2 " }}{PARA 4 "" 0 "
" {TEXT 274 16 "o simplificando " }}{PARA 4 "" 0 "" {TEXT 275 38 "    \+
  24 x^3 + 288 x + 24 x y^2 = 0   " }}{PARA 4 "" 0 "" {TEXT 276 35 "  \+
    x (24 x^2 + 288 + 24 y^2) = 0" }}{PARA 4 "" 0 "" {TEXT 277 94 "y l
a curva frontera del dominio se compone s\363lo de los puntos con  x =
 0, o sea, los z = i y, " }}{PARA 4 "" 0 "" {TEXT 278 231 "ya que la e
xpresi\363n  24 x^2 + 288 + 24 y^2  no puede anularse.\nQueda por dete
rminar cu\341l de los dos semiplanos\n      Re(z) < 0  o  Re(z) > 0\ns
eparados por la curva que hemos calculado, es el dominio de estabilida
d. Pues bien, como " }}{PARA 4 "" 0 "" {TEXT 279 19 "   r (-3) = 3/39 \+
, " }}{PARA 4 "" 0 "" {TEXT 280 120 "-3 est\341 en  D , y  D  coincide
 exactamente con el semiplano  Re(z) < 0 , por lo que el\nm\351todo de
l enunciado es A-estable" }{TEXT -1 0 "" }}}}{MARK "0 3 0" 15 }
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