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{SECT 0 {EXCHG {PARA 4 "" 0 "" {TEXT 273 14 "Ejercicios de " }}{PARA 
4 "" 0 "" {TEXT 274 22 "(8) Estabilidad lineal" }}{PARA 4 "" 0 "" 
{TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 272 51 "Ejercicio 08-09   (del E
XAMEN ORDINARIO de 07FEB03)" }}{PARA 4 "" 0 "" {TEXT -1 86 "\nConsid
\351rese el m\351todo gen\351rico expl\355cito de Runge-Kutta dado por
 el siguiente tablero" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 
"" {TEXT 256 13 "       0   | " }}{PARA 4 "" 0 "" {TEXT 257 6 "      \+
" }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 263 8 "   |    \+
" }{XPPEDIT 18 0 "a[21]" "6#&%\"aG6#\"#@" }{TEXT 264 2 "  " }}{PARA 4 
"" 0 "" {TEXT 265 6 "      " }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 266 
16 "   |      0     " }{XPPEDIT 18 0 "a[32]" "6#&%\"aG6#\"#K" }{TEXT 
267 2 "  " }}{PARA 4 "" 0 "" {TEXT -1 12 "   ---------" }{TEXT 258 24 
"------------------------" }}{PARA 4 "" 0 "" {TEXT 259 35 "           \+
 |       0      0      1" }}{PARA 4 "" 0 "" {TEXT -1 92 "a) D\355gase \+
cual es el orden m\341ximo que se puede alcanzar y cu\341les deben ser
 los coeficientes " }}{PARA 4 "" 0 "" {TEXT -1 25 "para alcanzar dicho
 orden" }}{PARA 4 "" 0 "" {TEXT -1 79 "b) De entre los m\351todos de o
rden m\341ximo, d\355gase cu\341les son los que minimizan el" }}{PARA 
4 "" 0 "" {TEXT -1 19 "error de truncaci\363n" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 46 "(lo que antecede es  Ejer
cicio del cap\355tulo 3)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "
" 0 "" {TEXT -1 91 "c) Para la familia de m\351todos propuesta, calc
\372lese la funci\363n de estabilidad. Calc\372lese la " }}{PARA 4 "" 
0 "" {TEXT -1 80 "misma funci\363n para los m\351todos de orden m\341x
imo y para los que minimizan el error" }}}{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:=s
ubs(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:=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 35 "k2_:=f(x0+(1/2)*h,y0+a[
2,1]*h*k1_):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "k3_:=f(x0+h
,y0+a[3,2]*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:=su
bs(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 "" {TEXT 262 2 "a)" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 260 27 "condiciones para el orden 1" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 19 "Fi_0:=expand(k3_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 
0 "" 0 "" {TEXT 275 27 "condiciones para el orden 2" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 19 "Fi10:=expand(k310);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%Fi10G,&--&%\"DG6#\"\"\"6#%\"fG6$%#x0G%#y0GF+*(--&F)6
#\"\"#F,F.F+&%\"aG6$\"\"$F6F+-F-F.F+F+" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 66 "cond2a:=(1/2)*coeff(Y2,D[1](f)(x0,y0))=coeff(Fi10,D[1
](f)(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond2aG/#\"\"\"\"
\"#F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "cond2b:=(1/2)*coef
f(coeff(Y2,D[2](f)(x0,y0)),f(x0,y0))=coeff(coeff(Fi10,D[2](f)(x0,y0)),
f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond2bG/#\"\"\"\"\"#&
%\"aG6$\"\"$F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 95 "y no se puede \+
conseguir el orden 2 de ninguna manera. El orden m\341ximo es 1 y cual
quiera de los " }}{PARA 0 "" 0 "" {TEXT 277 39 "m\351todos de la famil
ia tiene ese orden  " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 276 2 "b)" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 261 90 "Para minimizar el error, de entre
 los m\351todos de orden 1 (que son todos) habr\341 que elegir " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "sol1:=solve(\{cond2b\},\{a[3
,2]\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sol1G<#/&%\"aG6$\"\"$\"
\"##\"\"\"F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "a[3,2]:=sub
s(sol1,a[3,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6$\"\"$\"\"#
#\"\"\"F(" }}}{EXCHG {PARA 4 "" 0 "" {TEXT 278 14 "y se obtiene  " }
{XPPEDIT 18 0 "a[32]" "6#&%\"aG6#\"#K" }{TEXT 279 3 " = " }{XPPEDIT 
18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 280 57 "  como valor para l
a minimizaci\363n del error, siendo a\372n  " }{XPPEDIT 18 0 "a[21]" "
6#&%\"aG6#\"#@" }{TEXT 281 12 "  arbitrario" }{TEXT 282 1 " " }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 269 2 "c)" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 270 56 "funci\363n de estabilidad general y para el orden m\341x
imo 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 14 "k1_:=f(x0,y0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$
k1_G*&%'lambdaG\"\"\"%#y0GF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 35 "k2_:=f(x0+(1/2)*h,y0+a[2,1]*h*k1_);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$k2_G*&%'lambdaG\"\"\",&%#y0GF'**&%\"aG6$\"\"#F'F'%\"
hGF'F&F'F)F'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "k3_:=f(
x0+h,y0+a[3,2]*h*k2_);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$k3_G*&%'l
ambdaG\"\"\",&%#y0GF'**&%\"aG6$\"\"$\"\"#F'%\"hGF'F&F',&F)F'**&F,6$F/F
'F'F0F'F&F'F)F'F'F'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "
y1:=expand(y0+h*(k3_));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G,*%#y
0G\"\"\"*(%\"hGF'%'lambdaGF'F&F'F'**)F)\"\"#F')F*F-F'&%\"aG6$\"\"$F-F'
F&F'F'*,)F)F2F')F*F2F'F/F'&F06$F-F'F'F&F'F'" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 36 "y1:=collect(subs(lambda=z/h,y1),y0);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#y1G*&,*\"\"\"F'%\"zGF'*&)F(\"\"#F'&%\"aG6$\"
\"$F+F'F'*()F(F/F'F,F'&F-6$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$%)operatorG%&arrow
GF(,*\"\"\"F-9$F-*&)F.\"\"#F-&%\"aG6$\"\"$F1F-F-*()F.F5F-F2F-&F36$F1F-
F-F-F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 271 63 "funci\363n de esta
bilidad para los metodos que minimizan el error " }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 37 "r2:=unapply(subs(a[3,2]=1/2,r(z)),z);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r2Gf*6#%\"zG6\"6$%)operatorG%&arrow
GF(,*\"\"\"F-9$F-*&#F-\"\"#F-*$)F.F1F-F-F-*&F0F-*&)F.\"\"$F-&%\"aG6$F1
F-F-F-F-F(F(F(" }}}}{MARK "0 3 0" 15 }{VIEWOPTS 1 1 0 3 4 1802 1 1 1 
1 }{PAGENUMBERS 0 1 2 33 1 1 }