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{SECT 0 {EXCHG {PARA 4 "" 0 "" {TEXT 270 14 "Ejercicios de " }}{PARA 
4 "" 0 "" {TEXT 271 55 "(7) M\351todos de RUNGE-KUTTA impl\355citos y \+
semi-impl\355citos" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" 
{TEXT 269 15 "Ejercicio 07-01" }}{PARA 4 "" 0 "" {TEXT -1 84 "\na) B
\372squense los m\351todos con el mayor orden posible para la familia \+
de RUNGE-KUTTA " }}{PARA 4 "" 0 "" {TEXT -1 32 "impl\355citos del sigu
iente tablero" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" 
{TEXT 256 18 "     1    |      1" }}{PARA 4 "" 0 "" {TEXT 257 5 "     \+
" }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" }{TEXT 265 18 "   |     0   \+
     " }{XPPEDIT 18 0 "a[22]" "6#&%\"aG6#\"#A" }}{PARA 4 "" 0 "" 
{TEXT -1 12 "   ---------" }{TEXT 258 24 "------------------------" }}
{PARA 4 "" 0 "" {TEXT 259 18 "           |      " }{XPPEDIT 18 0 "b[1]
" "6#&%\"bG6#\"\"\"" }{TEXT 266 8 "        " }{XPPEDIT 18 0 "b[2]" "6#
&%\"bG6#\"\"#" }{TEXT 267 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 4 "" 0 "" {TEXT -1 68 "b) Pru\351bese que, entre los m\351todos \+
obtenidos, se encuentra el m\351todo" }}{PARA 4 "" 0 "" {TEXT -1 24 "d
e GAUSS de 1 sola etapa" }}{PARA 4 "" 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 68 "c) Para  \+
los m\351todos obtenidos,  calc\372lese la funci\363n de estabilidad" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 71 "d) Calc
\372lese la funci\363n de estabilidad para el citado m\351todo de GAUS
S, " }}{PARA 4 "" 0 "" {TEXT -1 30 "y compru\351bese que es A-estable
" }}}{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:=sub
s(\{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 37 "k1_:=unapply(f(x0+h,y0+h*kp1_(h)),h):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "k2_:=unapply(f(x0+c[2]*h,y0+a[2,2]*
h*kp2_(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,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(k
p1_)(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 34 "Fi_0:=expand(b[1]*k1_0+b[2]*k2_0);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%Fi_0G,&*&&%\"bG6#\"\"\"F*-%\"fG6$%#x0G%#y0GF*F**&&F(
6#\"\"#F*F+F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "cond1a:=
coeff(Y1,f(x0,y0))=coeff(Fi_0,f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%'cond1aG/\"\"\",&&%\"bG6#F&F&&F)6#\"\"#F&" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 14 "solve(cond1a);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#<$/&%\"bG6#\"\"#,&\"\"\"F*&F&6#F*!\"\"/F+F+" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT 261 27 "condiciones para el orden 2" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 34 "Fi10:=expand(b[1]*k110+b[2]*k210);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%%Fi10G,**&&%\"bG6#\"\"\"F*--&%\"DGF)6#%\"f
G6$%#x0G%#y0GF*F**(F'F*--&F.6#\"\"#F/F1F*-F0F1F*F**(&F(F8F*F+F*&%\"cGF
8F*F***F<F*F5F*&%\"aG6$F9F9F*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/#\"\"\"\"
\"#,&&%\"bG6#F'F'*&&F+6#F(F'&%\"cGF/F'F'" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 98 "cond2b:=(1/2)*coeff(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/#\"\"\"\"\"#,&&%\"bG6#F'F'*&&F+6#F(F'&%\"aG6
$F(F(F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "solucion:=solv
e(\{cond1a,cond2a,cond2b\},\{b[1],c[2],a[2,2]\});" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%)solucionG<%/&%\"cG6#\"\"#,$*&#\"\"\"F*F.*&,&F.!\"\"*
&F*F.&%\"bGF)F.F.F.F3F1F.F./&F46#F.,&F.F.F3F1/&%\"aG6$F*F*F+" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 262 28 "condiciones para el orden 3 " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Fi20:=expand(b[1]*k120+b[2]*
k220);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%Fi20G,6*&&%\"bG6#\"\"\"F*
--&%\"DG6$F*F*6#%\"fG6$%#x0G%#y0GF*F***\"\"#F*F'F*--&F.6$F*F6F0F2F*-F1
F2F*F**(F'F*--&F.6$F6F6F0F2F*)F;F6F*F***F6F*F'F*--&F.6#F6F0F2F*--&F.F)
F0F2F*F***F6F*F'F*)FCF6F*F;F*F**(&F(FFF*F+F*)&%\"cGFFF6F*F**.F6F*FMF*F
OF*F7F*&%\"aGF@F*F;F*F***FMF*)FRF6F*FAF*F=F*F**.F6F*FMF*FCF*FRF*FGF*FO
F*F**,F6F*FMF*FKF*FUF*F;F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 76 "cond3a:=(1/6)*coeff(Y3,D[1,1](f)(x0,y0))=(1/2)*coeff(Fi20,D[1,1]
(f)(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3aG/#\"\"\"\"\"
',&*&#F'\"\"#F'&%\"bG6#F'F'F'*&F+F'*&&F.6#F,F')&%\"cGF3F,F'F'F'" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "cond3b:=(1/6)*coeff(coeff(Y
3,D[1,2](f)(x0,y0)),f(x0,y0))=(1/2)*coeff(coeff(Fi20,D[1,2](f)(x0,y0))
,f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3bG/#\"\"\"\"\"$
,&&%\"bG6#F'F'*(&F+6#\"\"#F'&%\"cGF/F'&%\"aG6$F0F0F'F'" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "cond3c:=(1/6)*coeff(coeff(Y3,D[2,2
](f)(x0,y0)),f(x0,y0)^2)=(1/2)*coeff(coeff(Fi20,D[2,2](f)(x0,y0)),f(x0
,y0)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3cG/#\"\"\"\"\"',&*
&#F'\"\"#F'&%\"bG6#F'F'F'*&F+F'*&&F.6#F,F')&%\"aG6$F,F,F,F'F'F'" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "cond3d:=(1/6)*coeff(coeff(Y
3,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/#\"
\"\"\"\"',&&%\"bG6#F'F'*(&F+6#\"\"#F'&%\"cGF/F'&%\"aG6$F0F0F'F'" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "cond3e:=(1/6)*coeff(coeff(Y
3,D[2](f)(x0,y0)^2),f(x0,y0))=(1/2)*coeff(coeff(Fi20,D[2](f)(x0,y0)^2)
,f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3eG/#\"\"\"\"\"'
,&&%\"bG6#F'F'*&&F+6#\"\"#F')&%\"aG6$F0F0F0F'F'" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 65 "solve(\{cond1a,cond2a,cond2b,cond3a,cond3b,con
d3c,cond3d,cond3e\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 272 93 "Ni siqu
iera es posible igualar simult\341neamente  cond3a  y  cond3d , por ej
emplo, si se acepta" }}{PARA 0 "" 0 "" {TEXT 273 6 "que   " }{XPPEDIT 
18 0 "c[2]" "6#&%\"cG6#\"\"#" }{TEXT 274 5 "  =  " }{XPPEDIT 18 0 "a[2
2]" "6#&%\"aG6#\"#A" }{TEXT 275 45 "  , como se desprende de cond2a  y
  cond2b   " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "solve(\{c[2]
=a[2,2],cond3a,cond3d\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 2 "b)" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "solucion1:=subs(b[2]=1,so
lucion) union \{b[2]=1\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*soluci
on1G<&/&%\"bG6#\"\"\"\"\"!/&F(6#\"\"#F*/&%\"cGF.#F*F//&%\"aG6$F/F/F3" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT 277 43 "La soluci\363n obtenida corres
ponde al tablero" }}{PARA 4 "" 0 "" {TEXT 278 19 "      1    |      1
" }}{PARA 4 "" 0 "" {TEXT 279 5 "     " }{XPPEDIT 18 0 "1/2" "6#*&\"\"
\"F$\"\"#!\"\"" }{TEXT 282 19 "    |     0        " }{XPPEDIT 18 0 "1/
2" "6#*&\"\"\"F$\"\"#!\"\"" }}{PARA 4 "" 0 "" {TEXT -1 12 "   --------
-" }{TEXT 280 24 "------------------------" }}{PARA 4 "" 0 "" {TEXT 
281 30 "            |      0        1 " }}{PARA 0 "" 0 "" {TEXT 283 
72 "que no hace intervenir la primera etapa y que coincide, por lo tan
to con" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 
0 "" {TEXT 284 5 "     " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"
" }{TEXT 287 10 "    |     " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!
\"\"" }}{PARA 4 "" 0 "" {TEXT -1 12 "   ---------" }{TEXT 285 24 "----
--------------------" }}{PARA 4 "" 0 "" {TEXT 286 21 "            |   \+
   1 " }}{PARA 0 "" 0 "" {TEXT 288 44 "que no es sino el m\351todo de \+
Gauss de 1 etapa" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 276 2 "c)" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 268 23 "funci\363n de estabilidad " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f:=(x,y)->lambda*y;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"yG6\"6$%)operatorG%&
arrowGF)*&%'lambdaG\"\"\"9%F/F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 36 "ka1:=unapply(f(x0+h,y0+h*kp1(h)),h);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%$ka1Gf*6#%\"hG6\"6$%)operatorG%&arrowGF(*&%'lamb
daG\"\"\",&%#y0GF.*&9$F.-%$kp1G6#F2F.F.F.F(F(F(" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 48 "ka2:=unapply(f(x0+c[2]*h,y0+a[2,2]*h*kp2(h)),h
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ka2Gf*6#%\"hG6\"6$%)operatorG
%&arrowGF(*&%'lambdaG\"\"\",&%#y0GF.*(&%\"aG6$\"\"#F5F.9$F.-%$kp2G6#F6
F.F.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "test:=solve
(\{k1=subs(kp1(h)=k1,ka1(h)),k2=subs(kp2(h)=k2,ka2(h))\},\{k1,k2\});" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%testG<$/%#k2G,$*(%'lambdaG\"\"\"%
#y0GF+,&F+!\"\"*(F*F+&%\"aG6$\"\"#F3F+%\"hGF+F+F.F./%#k1G,$*(F*F+F,F+,
&F+F.*&F*F+F4F+F+F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "k1
:=subs(test,k1);k2:=subs(test,k2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%#k1G,$*(%'lambdaG\"\"\"%#y0GF(,&F(!\"\"*&F'F(%\"hGF(F(F+F+" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#k2G,$*(%'lambdaG\"\"\"%#y0GF(,&F(!\"\"*(F
'F(&%\"aG6$\"\"#F0F(%\"hGF(F(F+F+" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "y1:=expand(y0+h*(b[1]*k1+b[2]*k2));" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%#y1G,(%#y0G\"\"\"*,%\"hGF'&%\"bG6#F'F'%'lambdaGF'F
&F',&F'!\"\"*&F-F'F)F'F'F/F/*,F)F'&F+6#\"\"#F'F-F'F&F',&F'F/*(F-F'&%\"
aG6$F4F4F'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'*(&%\"bG6#F'F'%\"zGF',&F'!\"\"F,F'F.F.*(&F*6#\"\"#F'F
,F',&F'F.*&F,F'&%\"aG6$F2F2F'F'F.F.F'%#y0GF'" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 41 "r:=unapply(collect(simplify(y1/y0),z),z);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rGf*6#%\"zG6\"6$%)operatorG%&arrow
GF(,$*(,(\"\"\"!\"\"*&,(&%\"aG6$\"\"#F6F0*&&%\"bG6#F/F/F3F/F/&F96#F6F/
F/)9$F6F/F/*&,*F3F/F/F/F8F0F;F0F/F>F/F/F/,&F/F0F>F/F0,&F/F0*&F>F/F3F/F
/F0F0F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 290 64 "que es la funci
\363n de estabilidad de la familia general propuesta" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT 289 2 "d)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 291 79 "Par
a el m\351todo de GAUSS de 1 etapa, obtenido en el conjunto-solucion  \+
solucion1" }}{PARA 0 "" 0 "" {TEXT 292 35 "tenemos como funci\363n de \+
estabilidad" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "r1:=unapply(
simplify(subs(solucion1,r(z))),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%#r1Gf*6#%\"zG6\"6$%)operatorG%&arrowGF(,$*&,&9$\"\"\"\"\"#F0F0,&F1!
\"\"F/F0F3F3F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 293 50 "El dominio
 de estabilidad, compuesto por los  z = " }{XPPEDIT 18 0 "lambda" "6#%
'lambdaG" }{TEXT 294 35 "  h  tales que el m\363dulo de  r1(z) " }}
{PARA 0 "" 0 "" {TEXT 295 88 "sea inferior a  1 , estar\341 formado po
r los valores de  z  que disten menos del punto  -2" }}{PARA 0 "" 0 "
" {TEXT 296 86 "que del punto  2 . Esto corresponde al semiplano de pa
rte real negativa, por lo que el" }}{PARA 0 "" 0 "" {TEXT 297 30 "m
\351todo estudiado es  A-estable" }}}}{MARK "0 0 0" 2 }{VIEWOPTS 1 1 
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