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{SECT 0 {EXCHG {PARA 4 "" 0 "" {TEXT 263 14 "Ejercicios de " }}{PARA 
4 "" 0 "" {TEXT 264 22 "(8) Estabilidad lineal" }}{PARA 4 "" 0 "" 
{TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 262 55 "Ejercicio 08-08  (del EX
AMEN EXTRAORDINARIO de 09JUL03)" }}{PARA 4 "" 0 "" {TEXT -1 81 "\nCons
id\351rese la familia de m\351todos de RUNGE-KUTTA de 3 etapas y expl
\355citos dados " }}{PARA 4 "" 0 "" {TEXT -1 24 "por el siguiente tabl
ero" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 265 12 " \+
      0  | " }}{PARA 4 "" 0 "" {TEXT 266 15 "    1/2  |     " }
{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }}{PARA 4 "" 0 "" {TEXT 269 16 "  \+
     1  |     " }{XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT 270 7 "     \+
 2" }}{PARA 4 "" 0 "" {TEXT -1 12 "   ---------" }{TEXT 267 20 "------
--------------" }}{PARA 4 "" 0 "" {TEXT 268 36 "           |  1/6     \+
  2/3      1/6" }{TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 38 "a) Para \+
qu\351 valores de los par\341metros " }{XPPEDIT 18 0 "alpha" "6#%&alph
aG" }{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) E
xiste 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 4 "" 0 "" {TEXT -1 77 "\nc) Para la familia de m
\351todos propuesta, calc\372lese la funci\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 77 "d) Calc\372lese ahora l
a funci\363n de estabilidad para los valores para los que se" }}{PARA 
4 "" 0 "" {TEXT -1 46 "obtiene el orden 2 . Si en este caso hacemos  \+
" }{XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT -1 9 " = 1  o  " }
{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 23 " = 0 , que es lo mis
mo," }}{PARA 4 "" 0 "" {TEXT -1 51 "dib\372jese la correspondiente reg
i\363n de estabilidad. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "r
estart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "interface(labeli
ng=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:=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 32 "k2_:=f(x0+1/2*h,y0+alpha*h*k1_):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "k3_:=f(x0+h,y0+h*beta*k1_+h*2*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 259 2 "a)" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 256 27 "condiciones para el orden 1" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 41 "Fi_0:=expand(1/6*k1_0+2/3*k2_0+1/6*k3_0);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%%Fi_0G-%\"fG6$%#x0G%#y0G" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 48 "cond1a:=coeff(Y1,f(x0,y0))=coeff(Fi_0,f(x
0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond1aG/\"\"\"F&" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 257 27 "condiciones para el orden 2" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Fi10:=expand(1/6*k110+2/3*k2
10+1/6*k310);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Fi10G,**&#\"\"\"\"
\"#F(--&%\"DG6#F(6#%\"fG6$%#x0G%#y0GF(F(*&#F)\"\"$F(*(--&F-6#F)F/F1F(%
&alphaGF(-F0F1F(F(F(*&#F(\"\"'F(*(F8F(%%betaGF(F=F(F(F(*&#F(F6F(*&F8F(
F=F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "cond2a:=(1/2)*c
oeff(Y2,D[1](f)(x0,y0))=coeff(Fi10,D[1](f)(x0,y0));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%'cond2aG/#\"\"\"\"\"#F&" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 98 "cond2b:=(1/2)*coeff(coeff(Y2,D[2](f)(x0,y0)),f(x0,y
0))=coeff(coeff(Fi10,D[2](f)(x0,y0)),f(x0,y0));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'cond2bG/#\"\"\"\"\"#,(*(F(F'\"\"$!\"\"%&alphaGF'F'*&
\"\"'F,%%betaGF'F'#F'F+F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
36 "sol2:=solve(\{cond2a,cond2b\},\{beta\});" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%sol2G<#/%%betaG,&\"\"\"F)*&\"\"%F)%&alphaGF)!\"\"" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "beta:=subs(sol2,beta);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG,&\"\"\"F&*&\"\"%F&%&alphaGF&!
\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 2 "b)" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT 258 28 "condiciones para el orden 3 " }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 41 "Fi20:=expand(1/6*k120+2/3*k220+1/6*k320);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Fi20G,2*&#\"\"\"\"\"$F(--&%\"DG6$F(
F(6#%\"fG6$%#x0G%#y0GF(F(*&#\"\"#F)F(*(--&F-6$F(F6F/F1F(%&alphaGF(-F0F
1F(F(!\"\"*&#\"#5F)F(*()F<F6F()F=F6F(--&F-6$F6F6F/F1F(F(F(*&F8F(F=F(F(
*&#F)F6F(*&FEF(FDF(F(F(**\"\"%F(FEF(FDF(F<F(F>*&F'F(*&--&F-6#F6F/F1F(-
-&F-6#F(F/F1F(F(F(*&#F6F)F(*()FQF6F(F<F(F=F(F(F(" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 76 "cond3a:=(1/6)*coeff(Y3,D[1,1](f)(x0,y0))=(1/2
)*coeff(Fi20,D[1,1](f)(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'
cond3aG/#\"\"\"\"\"'F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "
cond3b:=(1/6)*coeff(coeff(Y3,D[1,2](f)(x0,y0)),f(x0,y0))=(1/2)*coeff(c
oeff(Fi20,D[1,2](f)(x0,y0)),f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%'cond3bG/#\"\"\"\"\"$,&*&F(!\"\"%&alphaGF'F+#F'\"\"#F'" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "cond3c:=(1/6)*coeff(coeff(Y
3,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/#\"\"\"
\"\"',(*&#\"\"&\"\"$F'*$)%&alphaG\"\"#F'F'F'#F-\"\"%F'*&F1F'F0F'!\"\"
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "cond3d:=(1/6)*coeff(co
eff(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&" }}}{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(Fi
20,D[2](f)(x0,y0)^2),f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'
cond3eG/#\"\"\"\"\"',$*&\"\"$!\"\"%&alphaGF'F'" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 51 "sol3:=solve(\{cond3b,cond3c,cond3d,cond3e\},\{
alpha\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sol3G<#/%&alphaG#\"\"
\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "alpha:=subs(sol3
,alpha);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG#\"\"\"\"\"#" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 271 2 "c)" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 261 37 "funci\363n de estabilidad, caso general " }}}{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)*&%'lambd
aG\"\"\"9%F/F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "k1_:=
f(x0,y0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$k1_G*&%'lambdaG\"\"\"%
#y0GF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "k2_:=f(x0+1/2*h,y
0+alpha*h*k1_);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$k2_G*&%'lambdaG
\"\"\",&%#y0GF'**%&alphaGF'%\"hGF'F&F'F)F'F'F'" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 35 "k3_:=f(x0+h,y0+h*beta*k1_+h*2*k2_);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%$k3_G*&%'lambdaG\"\"\",(%#y0GF'**%\"hGF'%%
betaGF'F&F'F)F'F'**\"\"#F'F+F'F&F',&F)F'**%&alphaGF'F+F'F&F'F)F'F'F'F'
F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "y1:=expand(y0+h*(1/6*
k1_+2/3*k2_+1/6*k3_));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G,.%#y0
G\"\"\"*(%\"hGF'%'lambdaGF'F&F'F'*.\"\"#F'\"\"$!\"\"F)F,F*F,%&alphaGF'
F&F'F'*,\"\"'F.F)F,%%betaGF'F*F,F&F'F'**F-F.F)F,F*F,F&F'F'*,F-F.F)F-F*
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'*&#\"\"#\"\"$F'*&)F(F+F'%&alphaGF'F'F'*&#F'\"\"'F'*&%%
betaGF'F.F'F'F'*&#F'F,F'*$F.F'F'F'*&F6F'*&)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(,*\"\"\"F-*&#F-\"\"$F-*&)9$F0F-%&alphaGF-F-F-*&,(
*(\"\"#F-F0!\"\"F4F-F-*&\"\"'F9%%betaGF-F-F/F-F-)F3F8F-F-F3F-F(F(F(" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT 273 2 "d)" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 272 32 "funci\363n de estabilidad, orden 2 " }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 18 "beta := 1-4*alpha;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%betaG,&\"\"\"F&*&\"\"%F&%&alphaGF&!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "r(z);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,*\"\"\"F$*&#F$\"\"$F$*&)%\"zGF'F$%&alphaGF$F$F$*&#F$\"\"#F$*$)F
*F.F$F$F$F*F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 274 28 "regi\363n de es
tabilidad para  " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT 275 6 " \+
= 0  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "alpha:=0;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%&alphaG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 5 "r(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(\"\"\"F$*&#
F$\"\"#F$*$)%\"zGF'F$F$F$F*F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 276 54 
"la regi\363n de estabilidad es la figura ya estudiada:   " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "z:=x+I*y:with(plots):implicitplot(
abs(r(z))=1,x=-3..1,y=-3..3,scaling=constrained,axesfont=[COURIER,BOLD
,18]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CUR
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