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{SECT 0 {EXCHG {PARA 4 "" 0 "" {TEXT 285 14 "Ejercicios de " }}{PARA 
4 "" 0 "" {TEXT 286 22 "(8) Estabilidad lineal" }}{PARA 4 "" 0 "" 
{TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 267 50 "Ejercicio 08-16  (del EX
AMEN ORDINARIO de 09FEB04)" }}{PARA 4 "" 0 "" {TEXT -1 65 "\nPara los \+
m\351todos de RUNGE-KUTTA impl\355citos del siguiente tablero" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 256 5 "     " }
{XPPEDIT 18 0 "1/3;" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT 272 11 "    |    \+
  " }{XPPEDIT 18 0 "1/3" "6#*&\"\"\"F$\"\"$!\"\"" }}{PARA 4 "" 0 "" 
{TEXT 257 18 "     1    |       " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG
" }{TEXT 273 10 "         0" }}{PARA 4 "" 0 "" {TEXT -1 12 "   -------
--" }{TEXT 258 15 "---------------" }}{PARA 4 "" 0 "" {TEXT 259 18 "  \+
         |      " }{XPPEDIT 18 0 "3/4;" "6#*&\"\"$\"\"\"\"\"%!\"\"" }
{TEXT 264 9 "         " }{XPPEDIT 18 0 "1/4;" "6#*&\"\"\"F$\"\"%!\"\"
" }{TEXT 265 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" 
{TEXT -1 25 "a) d\355gase qu\351 valores de " }{XPPEDIT 18 0 "alpha;" 
"6#%&alphaG" }{TEXT -1 35 "  proporcionan m\351todos de orden 2 ;" }}
{PARA 4 "" 0 "" {TEXT -1 32 "b) d\355gase si existen valores de " }
{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 39 "  que proporcionen m
\351todos de orden 3 ;" }}{PARA 4 "" 0 "" {TEXT -1 32 "c) d\355gase si
 existen valores de " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 
40 "  que proporcionen m\351todos de orden 4 . " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 46 "(lo que antecede es  Ejer
cicio del cap\355tulo 7)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "
" 0 "" {TEXT -1 73 "d) Calc\372lese la funci\363n de estabilidad para \+
 los m\351todos de la familia.  " }}{PARA 4 "" 0 "" {TEXT -1 67 "e) Di
sc\372tase sobre la  A-estabilidad de los m\351todos de la familia. " 
}}{PARA 4 "" 0 "" {TEXT -1 78 "f) Calc\372lese la funci\363n de estabi
lidad para  los m\351todos obtenidos en b) y c)." }}}{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:=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:=sub
s(\{y(x)=y0,x=x0\},y3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "
k1_:=unapply(f(x0+(1/3)*h,y0+(1/3)*h*kp1_(h)),h):" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 43 "k2_:=unapply(f(x0+h,y0+alpha*h*kp1_(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 "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,k
p2_(0)=k2_0\},k21(0)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "k
12:=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 "condicio
nes para el orden 1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Fi_0
:=expand((3/4)*k1_0+(1/4)*k2_0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
%Fi_0G-%\"fG6$%#x0G%#y0G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 
"cond1a:=coeff(Y1,f(x0,y0))=coeff(Fi_0,f(x0,y0));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'cond1aG/\"\"\"F&" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 
-1 33 "El orden es siempre de 1 al menos" }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 261 27 "condiciones para el orden 2" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Fi10:=expand((3/4)*k110+(1/4
)*k210);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Fi10G,(*&#\"\"\"\"\"#F(
--&%\"DG6#F(6#%\"fG6$%#x0G%#y0GF(F(*&#F(\"\"%F(*&--&F-6#F)F/F1F(-F0F1F
(F(F(*&F5F(*(F8F(%&alphaGF(F<F(F(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&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "cond2b:=(1/2)*coef
f(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'*&F+!\"\"%&alphaGF'F'" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "solucion:=solve(\{cond2a,cond2b\});" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%)solucionG<#/%&alphaG\"\"\"" }}}{EXCHG {PARA 256 "
" 0 "" {TEXT -1 9 "El valor " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }
{TEXT -1 43 " = 1   es el \372nico que proporciona orden 2 " }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 31 "b) condicion
es para el orden 3 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Fi20
:=expand((3/4)*k120+(1/4)*k220);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%
%Fi20G,4*&#\"\"\"\"\"$F(--&%\"DG6$F(F(6#%\"fG6$%#x0G%#y0GF(F(*&#F(\"\"
'F(*&--&F-6$F(\"\"#F/F1F(-F0F1F(F(F(*&#F(\"#7F(*&--&F-6$F<F<F/F1F()F=F
<F(F(F(*&F5F(*&--&F-6#F<F/F1F(--&F-6#F(F/F1F(F(F(*&F5F(*&)FIF<F(F=F(F(
F(*&#F(F<F(*(F8F(%&alphaGF(F=F(F(F(*&#F(\"\"%F(*()FWF<F(FFF(FBF(F(F(*&
F5F(*(FIF(FWF(FMF(F(F(*&F5F(*(FSF(FWF(F=F(F(F(" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 106 "cond3a:=(1/6)*coeff(Y3,D[1,1](f)(x0,y0))=(1/2
)*coeff(Fi20,D[1,1](f)(x0,y0)):cond3a:=subs(solucion,cond3a);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3aG/#\"\"\"\"\"'F&" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "cond3b:=(1/6)*coeff(coeff(Y3,D[1,2
](f)(x0,y0)),f(x0,y0))=(1/2)*coeff(coeff(Fi20,D[1,2](f)(x0,y0)),f(x0,y
0)):cond3b:=subs(solucion,cond3b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%'cond3bG/#\"\"\"\"\"$F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
142 "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):cond3c:=subs(solucion,c
ond3c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3cG/#\"\"\"\"\"'F&" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "cond3d:=(1/6)*coeff(coef
f(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)):cond3d:=subs(solucion,cond3d);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%'cond3dG/#\"\"\"\"\"'F&" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 138 "cond3e:=(1/6)*coeff(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)):cond3e:=sub
s(solucion,cond3e);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3eG/#\"
\"\"\"\"'F&" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 19 "Con el mismo val
or " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 25 " = 1 se obtien
e orden 3  " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 274 
11 "c) orden 4 " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 70 "En ning\372n \+
caso aparece en la familia el m\351todo de GAUSS de dos etapas." }}
{PARA 0 "" 0 "" {TEXT 275 44 "Por lo tanto, no se alcanza nunca el ord
en 4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 269 2 "d)" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT 266 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 48 "ka1:=
unapply(f(x0+(1/3)*h,y0+(1/3)*h*kp1(h)),h);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$ka1Gf*6#%\"hG6\"6$%)operatorG%&arrowGF(*&%'lambdaG\"
\"\",&%#y0GF.*&#F.\"\"$F.*&9$F.-%$kp1G6#F5F.F.F.F.F(F(F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "test:=solve(\{k1=subs(kp1(h)=k1,ka1
(h))\},k1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%testG<#/%#k1G,$**\"
\"$\"\"\"%'lambdaGF+%#y0GF+,&F*!\"\"*&F,F+%\"hGF+F+F/F/" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "k1:=subs(test,k1);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#k1G,$**\"\"$\"\"\"%'lambdaGF(%#y0GF(,&F'!\"\"*&
F)F(%\"hGF(F(F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "k2:=f(x0+h,y0+alpha*h*k1);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k2G*&%'lambdaG\"\"\",&%#y0GF'*.\"
\"$F'%&alphaGF'%\"hGF'F&F'F)F',&F+!\"\"*&F&F'F-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 37 "y1:=exp
and(y0+h*((3/4)*k1+(1/4)*k2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y
1G,*%#y0G\"\"\"*.\"\"*F'\"\"%!\"\"%\"hGF'%'lambdaGF'F&F',&\"\"$F+*&F-F
'F,F'F'F+F+**F*F+F,F'F-F'F&F'F'*0F/F'F*F+%&alphaGF'F,\"\"#F-F4F&F'F.F+
F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "y1:=collect(subs(lamb
da=z/h,y1),y0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G*&,*\"\"\"F'*
*\"\"*F'\"\"%!\"\"%\"zGF',&\"\"$F+F,F'F+F+*&F*F+F,F'F'*,F.F'F*F+%&alph
aGF'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$%)operatorG%&arrowGF(,$*&#\"\"\"\"
\"%F/*&,(\"#7F/*&,&F/!\"\"*&\"\"$F/%&alphaGF/F/F/)9$\"\"#F/F/*&\"\")F/
F;F/F/F/,&F8F6F;F/F6F/F6F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 271 
64 "que es la funci\363n de estabilidad de la familia general propuest
a" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 278 2 "e)" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT 279 30 "funci\363n de estabilidad cuando " }{XPPEDIT 18 0 "alp
ha;" "6#%&alphaG" }{TEXT 280 3 " = " }{XPPEDIT 18 0 "1/3;" "6#*&\"\"\"
F$\"\"$!\"\"" }{TEXT 281 35 " (\372nica posible A-estabilidad)     " }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "r_estab:=simplify(subs(\{a
lpha=1/3\},r(z)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(r_estabG,$*&,
&\"\"$\"\"\"*&\"\"#F)%\"zGF)F)F),&F(!\"\"F,F)F.F." }}}{EXCHG {PARA 0 "
" 0 "" {TEXT 282 43 "expresi\363n con l\355mite 2 cuando  z  tiende a \+
" }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT 283 28 " , por lo \+
que no puede haber" }}{PARA 0 "" 0 "" {TEXT 284 14 "A-estabilidad " }}
}{EXCHG {PARA 0 "" 0 "" {TEXT 270 2 "f)" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 276 30 "funci\363n de estabilidad cuando " }{XPPEDIT 18 0 "alpha
;" "6#%&alphaG" }{TEXT 277 26 " = 1 (orden m\341ximo 3)     " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "r_orden3:=simplify(subs(solu
cion,r(z)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)r_orden3G,$*(\"\"#!
\"\",(\"\"'\"\"\"*$)%\"zGF'F+F+*&\"\"%F+F.F+F+F+,&\"\"$F(F.F+F(F(" }}}
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