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{SECT 0 {EXCHG {PARA 4 "" 0 "" {TEXT 268 14 "Ejercicios de " }}{PARA 
4 "" 0 "" {TEXT 269 55 "(7) M\351todos de RUNGE-KUTTA impl\355citos y \+
semi-impl\355citos" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" 
{TEXT 267 50 "Ejercicio 07-04  (del EXAMEN 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 274 11 "    |      " }{XPPEDIT 18 0 "1/3" "6#*&\"
\"\"F$\"\"$!\"\"" }}{PARA 4 "" 0 "" {TEXT 257 18 "     1    |       " 
}{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT 275 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\355g
ase 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 "alph
a;" "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 proporcion
en m\351todos de orden 4 . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
4 "" 0 "" {TEXT -1 40 "(desde aqu\355 es Ejercicio del cap\355tulo 8)
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 73 "d) Ca
lc\372lese la funci\363n de estabilidad para  los m\351todos de la fam
ilia.  " }}{PARA 4 "" 0 "" {TEXT -1 67 "e) Disc\372tase sobre la  A-es
tabilidad de los m\351todos de la familia. " }}{PARA 4 "" 0 "" {TEXT 
-1 78 "f) Calc\372lese la funci\363n de estabilidad para  los m\351tod
os obtenidos en b) y c)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "
restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "interface(label
ing=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:=sub
s(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 49 "k1_:=unapply(f(x0+(1/3)*h,y0+(1/3)*
h*kp1_(h)),h):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "k2_:=unap
ply(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 "k1
1:=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,kp2_(0)=k2_0\},k21(0)):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "k12:=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 "condiciones 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$%#x0G%#y0G
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "cond1a:=coeff(Y1,f(x0,y
0))=coeff(Fi_0,f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond1a
G/\"\"\"F&" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 33 "El orden es siemp
re de 1 al menos" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
261 27 "condiciones para el orden 2" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 36 "Fi10:=expand((3/4)*k110+(1/4)*k210);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%%Fi10G,(*&#\"\"\"\"\"#F(--&%\"DG6#F(6#%\"fG6$%#x
0G%#y0GF(F(*&#F(\"\"%F(*&--&F-6#F)F/F1F(-F0F1F(F(F(*&F5F(*(F8F(%&alpha
GF(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)*coeff(coeff(Y2,D[2](f)(x0,y0)),f(x
0,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(\{co
nd2a,cond2b\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)solucionG<#/%&al
phaG\"\"\"" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 9 "El valor " }
{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 43 " = 1   es el \372nic
o que proporciona orden 2 " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT 262 31 "b) condiciones 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$F<F<F/F1F()F=F<F(F(F(*&F5F(*&--&F-6#F<F/F1F(--&F-6
#F(F/F1F(F(F(*&F5F(*&)FIF<F(F=F(F(F(*&#F(F<F(*(F8F(%&alphaGF(F=F(F(F(*
&#F(\"\"%F(*()FWF<F(FFF(FBF(F(F(*&F5F(*(FIF(FWF(FMF(F(F(*&F5F(*(FSF(FW
F(F=F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "cond3a:=(1/6
)*coeff(Y3,D[1,1](f)(x0,y0))=(1/2)*coeff(Fi20,D[1,1](f)(x0,y0)):cond3a
:=subs(solucion,cond3a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3aG
/#\"\"\"\"\"'F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "cond3b:
=(1/6)*coeff(coeff(Y3,D[1,2](f)(x0,y0)),f(x0,y0))=(1/2)*coeff(coeff(Fi
20,D[1,2](f)(x0,y0)),f(x0,y0)):cond3b:=subs(solucion,cond3b);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3bG/#\"\"\"\"\"$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,cond3c);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%'cond3cG/#\"\"\"\"\"'F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 146 "cond3d:=(1/6)*coeff(coeff(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(so
lucion,cond3d);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3dG/#\"\"\"
\"\"'F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "cond3e:=(1/6)*c
oeff(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:=subs(solucion,cond3e);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%'cond3eG/#\"\"\"\"\"'F&" }}}{EXCHG {PARA 256 "" 
0 "" {TEXT -1 19 "Con el mismo valor " }{XPPEDIT 18 0 "alpha;" "6#%&al
phaG" }{TEXT -1 25 " = 1 se obtiene orden 3  " }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 276 11 "c) orden 4 " }}}{EXCHG {PARA 0 "
" 0 "" {TEXT 270 70 "En ning\372n caso aparece en la familia el m\351t
odo de GAUSS de dos etapas." }}{PARA 0 "" 0 "" {TEXT 277 44 "Por lo ta
nto, no se alcanza nunca el orden 4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
271 2 "d)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 266 23 "funci\363n de estab
ilidad " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f:=(x,y)->lambda
*y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"yG6\"6$%)oper
atorG%&arrowGF)*&%'lambdaG\"\"\"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%&arr
owGF(*&%'lambdaG\"\"\",&%#y0GF.*&#F.\"\"$F.*&9$F.-%$kp1G6#F5F.F.F.F.F(
F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "test:=solve(\{k1=su
bs(kp1(h)=k1,ka1(h))\},k1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%test
G<#/%#k1G,$**\"\"$\"\"\"%'lambdaGF+%#y0GF+,&F*!\"\"*&F,F+%\"hGF+F+F/F/
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "k1:=subs(test,k1);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k1G,$**\"\"$\"\"\"%'lambdaGF(%#y0GF
(,&F'!\"\"*&F)F(%\"hGF(F(F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "k2:=f(x0+h,y0+alph
a*h*k1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k2G*&%'lambdaG\"\"\",&%
#y0GF'*.\"\"$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:=
expand(y0+h*((3/4)*k1+(1/4)*k2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%#y1G,*%#y0G\"\"\"*.\"\"*F'\"\"%!\"\"%\"hGF'%'lambdaGF'F&F',&\"\"$F+*&
F-F'F,F'F'F+F+**F*F+F,F'F-F'F&F'F'*0F/F'F*F+%&alphaGF'F,\"\"#F-F4F&F'F
.F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "y1:=collect(subs(l
ambda=z/h,y1),y0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G*&,*\"\"\"
F'**\"\"*F'\"\"%!\"\"%\"zGF',&\"\"$F+F,F'F+F+*&F*F+F,F'F'*,F.F'F*F+%&a
lphaGF'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 273 
64 "que es la funci\363n de estabilidad de la familia general propuest
a" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 280 2 "e)" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT 281 30 "funci\363n de estabilidad cuando " }{XPPEDIT 18 0 "alp
ha;" "6#%&alphaG" }{TEXT 282 3 " = " }{XPPEDIT 18 0 "1/3;" "6#*&\"\"\"
F$\"\"$!\"\"" }{TEXT 283 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 284 43 "expresi\363n con l\355mite 2 cuando  z  tiende a \+
" }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT 285 28 " , por lo \+
que no puede haber" }}{PARA 0 "" 0 "" {TEXT 286 14 "A-estabilidad " }}
}{EXCHG {PARA 0 "" 0 "" {TEXT 272 2 "f)" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 278 30 "funci\363n de estabilidad cuando " }{XPPEDIT 18 0 "alpha
;" "6#%&alphaG" }{TEXT 279 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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