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{SECT 0 {EXCHG {PARA 4 "" 0 "" {TEXT 270 14 "Ejercicios de " }}{PARA 
4 "" 0 "" {TEXT 272 22 "(8) Estabilidad lineal" }}{PARA 4 "" 0 "" 
{TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 271 16 "Ejercicio 08-02 " }}
{PARA 4 "" 0 "" {TEXT -1 81 "\nConsid\351rese la familia de m\351todos
 de Runge-Kutta de 3 etapas y expl\355citos dados " }}{PARA 4 "" 0 "" 
{TEXT -1 14 "por el tablero" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
4 "" 0 "" {TEXT 256 12 "       0  | " }}{PARA 4 "" 0 "" {TEXT 257 15 "
    3/2  |     " }{XPPEDIT 18 0 "beta" "6#%%betaG" }}{PARA 4 "" 0 "" 
{TEXT 260 23 "       0  |     1    -1" }}{PARA 4 "" 0 "" {TEXT -1 12 "
   ---------" }{TEXT 258 16 "----------------" }}{PARA 4 "" 0 "" 
{TEXT 259 22 "           |  1/3     " }{XPPEDIT 18 0 "alpha" "6#%&alph
aG" }{TEXT 261 8 "     1/3" }}{PARA 4 "" 0 "" {TEXT -1 38 "a) Para qu
\351 valores de los par\341metros " }{XPPEDIT 18 0 "alpha" "6#%&alphaG
" }{TEXT -1 4 " y  " }{XPPEDIT 18 0 "beta" "6#%%betaG" }{TEXT -1 29 " \+
los m\351todos son de orden 2 ?" }}{PARA 4 "" 0 "" {TEXT -1 29 "b) Exi
ste alguna elecci\363n de " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }
{TEXT -1 4 " y  " }{XPPEDIT 18 0 "beta" "6#%%betaG" }{TEXT -1 39 " que
 proporcione un m\351todo de orden 3 ?" }}{PARA 4 "" 0 "" {TEXT -1 0 "
" }}{PARA 4 "" 0 "" {TEXT -1 46 "(lo que antecede es  Ejercicio del ca
p\355tulo 3)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 
-1 76 "c) Para la familia de m\351todos propuesta, calc\372lese la fun
ci\363n de estabilidad" }}{PARA 4 "" 0 "" {TEXT -1 15 "en t\351rminos \+
de " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT -1 4 " y  " }{XPPEDIT 
18 0 "beta" "6#%%betaG" }{TEXT -1 2 " ." }}{PARA 4 "" 0 "" {TEXT -1 
26 "d) En el supuesto de que  " }{XPPEDIT 18 0 "abs(alpha)<2/3" "6#2-%
$absG6#%&alphaG*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 12 "  y de que  " }
{XPPEDIT 18 0 "beta=0" "6#/%%betaG\"\"!" }{TEXT -1 28 " , repres\351nt
ese la regi\363n de" }}{PARA 4 "" 0 "" {TEXT -1 24 "estabilidad del m
\351todo ." }}}{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 14 "k1_:=f(x0,y0):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "k2_:=f(x0+3/2*h,y0+beta*h*k1_):" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 26 "k3_:=f(x0,y0+h*k1_-h*k2_):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 20 "k1_0:=subs(h=0,k1_):" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 20 "k2_0:=subs(h=0,k2_):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 20 "k3_0:=subs(h=0,k3_):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 17 "k11:=diff(k1_,h):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "k21:=diff(k2_,h):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "k31:=diff(k3_,h):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "k110:=subs(h=0,k11):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "k210:=subs(h=0,k21):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "k310:=subs(h=0,k31):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "k12:=diff(k11,h):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "k22:=diff(k21,h):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "k32:=diff(k31,h):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "k120:=subs(h=0,k12):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "k220:=subs(h=0,k22):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "k320:=subs(h=0,k32):" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 265 2 "a)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 27 "condiciones p
ara el orden 1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Fi_0:=exp
and(1/3*k1_0+alpha*k2_0+1/3*k3_0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%%Fi_0G,&*&#\"\"#\"\"$\"\"\"-%\"fG6$%#x0G%#y0GF*F**&%&alphaGF*F+F*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/\"\"\",&#\"\"#\"\"$F&%&alphaGF&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "sol1:=solve(cond1a,\{alpha\});" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%sol1G<#/%&alphaG#\"\"\"\"\"$" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 24 "alpha:=subs(sol1,alpha);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&alphaG#\"\"\"\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
263 27 "condiciones para el orden 2" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 43 "Fi10:=expand(1/3*k110+alpha*k210+1/3*k310);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%%Fi10G,&*&#\"\"\"\"\"#F(--&%\"DG6#F(6#%\"f
G6$%#x0G%#y0GF(F(*&#F(\"\"$F(*(--&F-6#F)F/F1F(%%betaGF(-F0F1F(F(F(" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "cond2a:=(1/2)*coeff(Y2,D[1]
(f)(x0,y0))=coeff(Fi10,D[1](f)(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%'cond2aG/#\"\"\"\"\"#F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 98 "cond2b:=(1/2)*coeff(coeff(Y2,D[2](f)(x0,y0)),f(x0,y0))=coeff(c
oeff(Fi10,D[2](f)(x0,y0)),f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%'cond2bG/#\"\"\"\"\"#,$*&\"\"$!\"\"%%betaGF'F'" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 36 "sol2:=solve(\{cond2a,cond2b\},\{beta\});" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sol2G<#/%%betaG#\"\"$\"\"#" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "beta:=subs(sol2,beta);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG#\"\"$\"\"#" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT 266 2 "b)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 28 "con
diciones para el orden 3 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
43 "Fi20:=expand(1/3*k120+alpha*k220+1/3*k320);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%Fi20G,,*&#\"\"$\"\"%\"\"\"--&%\"DG6$F*F*6#%\"fG6$%#x
0G%#y0GF*F**&#F(\"\"#F**&--&F.6$F*F7F0F2F*-F1F2F*F*F**&F'F**&--&F.6$F7
F7F0F2F*)F=F7F*F*F**&--&F.6#F7F0F2F*--&F.6#F*F0F2F*!\"\"*&)FFF7F*F=F*F
N" }}}{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/#\"\"\"\"\"'#\"\"$\"\")" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "cond3b:=(1/6)*coeff(coeff(Y3,D[1,2
](f)(x0,y0)),f(x0,y0))=(1/2)*coeff(coeff(Fi20,D[1,2](f)(x0,y0)),f(x0,y
0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3bG/#\"\"\"\"\"$#F(\"\"
%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "cond3c:=(1/6)*coeff(c
oeff(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/#
\"\"\"\"\"'#\"\"$\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "
cond3d:=(1/6)*coeff(coeff(Y3,D[2](f)(x0,y0)),D[1](f)(x0,y0))=(1/2)*coe
ff(coeff(Fi20,D[2](f)(x0,y0)),D[1](f)(x0,y0));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'cond3dG/#\"\"\"\"\"'#!\"\"\"\"#" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 108 "cond3e:=(1/6)*coeff(coeff(Y3,D[2](f)(x0,y0)
^2),f(x0,y0))=(1/2)*coeff(coeff(Fi20,D[2](f)(x0,y0)^2),f(x0,y0));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3eG/#\"\"\"\"\"'#!\"\"\"\"#" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT 268 2 "c)" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 269 23 "funci\363n de estabilidad " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 
"interface(labeling=false):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
19 "f:=(x,y)->lambda*y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%
\"xG%\"yG6\"6$%)operatorG%&arrowGF)*&%'lambdaG\"\"\"9%F/F)F)F)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "k1_:=f(x0,y0);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%$k1_G*&%'lambdaG\"\"\"%#y0GF'" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "k2_:=f(x0+3/2*h,y0+beta*h*k1_);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$k2_G*&%'lambdaG\"\"\",&%#y0GF'**%%b
etaGF'%\"hGF'F&F'F)F'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
26 "k3_:=f(x0,y0+h*k1_-h*k2_);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$k
3_G*&%'lambdaG\"\"\",(%#y0GF'*(%\"hGF'F&F'F)F'F'*(F+F'F&F',&F)F'**%%be
taGF'F+F'F&F'F)F'F'F'!\"\"F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 45 "y1:=expand(y0+h*(1/3*k1_+alpha*k2_+1/3*k3_));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#y1G,,%#y0G\"\"\"*,\"\"#F'\"\"$!\"\"%\"hGF'%'lambda
GF'F&F'F'**F,F'%&alphaGF'F-F'F&F'F'*,F/F')F-F)F'%%betaGF')F,F)F'F&F'F'
*,F*F+F,F*F-F*F2F'F&F'F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 
"y1:=collect(subs(lambda=z/h,y1),y0);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%#y1G*&,,\"\"\"F'*&#\"\"#\"\"$F'%\"zGF'F'*&%&alphaGF'F,F'F'*(F.F')
F,F*F'%%betaGF'F'*&#F'F+F'*&)F,F+F'F1F'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$%)operat
orG%&arrowGF(,*\"\"\"F-*&#F-\"\"$F-*&)9$F0F-%%betaGF-F-!\"\"*(%&alphaG
F-)F3\"\"#F-F4F-F-*&,&#F9F0F-F7F-F-F3F-F-F(F(F(" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT 267 2 "d)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "beta
:=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG\"\"!" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "r(z);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,&\"\"\"F$*&,&#\"\"#\"\"$F$%&alphaGF$F$%\"zGF$F$" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "region:=abs(r(z))<1;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%'regionG2-%$absG6#,&\"\"\"F**&,&#\"\"#\"\"$F*%
&alphaGF*F*%\"zGF*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "d
iv:=2/3+alpha;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$divG,&#\"\"#\"\"$
\"\"\"%&alphaGF)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "region:
=abs(collect(r(z)/div,z))<1/div;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
'regionG2-%$absG6#,&%\"zG\"\"\"*&F+F+,&#\"\"#\"\"$F+%&alphaGF+!\"\"F+F
," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "radio:=1/div;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&radioG*&\"\"\"F&,&#\"\"#\"\"$F&%&al
phaGF&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "with(plots)
:alpha:=1/3;\nimplicitplot((x+radio)^2+y^2=radio^2,x=-5..5,y=-3..3\n  \+
           ,xtickmarks=[-1=`- ( 2 / 3 + a )`],ytickmarks=0,scaling=con
strained,axesfont=[COURIER,BOLD,18]);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%&alphaG#\"\"\"\"\"$" }}{PARA 13 "" 1 "" {GLPLOT2D 836 580 580 
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