{VERSION 6 0 "IBM INTEL NT" "6.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 236 0 76 0 0 0 2 0 0 
0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 48 37 0 0 0 0 0 0 0 0 0 0 
0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 0 0 0 0 1 }{CSTYLE "" -1 256 
"" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 1 14 0 0 0 
0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 1 }{CSTYLE "" -1 259 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 260 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
261 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 1 14 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 265 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
266 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 1 14 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 270 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
271 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 1 14 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 275 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
276 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 277 "" 1 14 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 278 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 279 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 280 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
281 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 282 "" 1 14 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 285 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 286 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 287 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
288 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 289 "" 1 14 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 290 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 291 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 292 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
293 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 294 "" 1 14 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 295 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 296 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 297 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
298 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 299 "" 1 14 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 300 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 301 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 302 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal
" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 
1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 
-1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 4 4 1 0 1 0 2 2 
0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 
0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }}
{SECT 0 {EXCHG {PARA 4 "" 0 "" {TEXT 301 14 "Ejercicios de " }}{PARA 
4 "" 0 "" {TEXT 302 55 "(7) M\351todos de RUNGE-KUTTA impl\355citos y \+
semi-impl\355citos" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" 
{TEXT 260 51 "Ejercicio 07-12   (del EXAMEN ORDINARIO DE 03JUL10)" }}
{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 75 "Consid
\351rese la familia de m\351todos impl\355citos de RUNGE-KUTTA con dos
 etapas " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 261 
5 "     " }{XPPEDIT 18 0 "0;" "6#\"\"!" }{TEXT 262 13 "       |     " 
}{XPPEDIT 18 0 "0;" "6#\"\"!" }{TEXT 263 2 "  " }}{PARA 4 "" 0 "" 
{TEXT 264 5 "     " }{XPPEDIT 18 0 "C;" "6#%\"CG" }{TEXT 265 11 "     \+
|     " }{XPPEDIT 18 0 "A[1];" "6#&%\"AG6#\"\"\"" }{TEXT 271 9 "      \+
   " }{XPPEDIT 18 0 "A[2];" "6#&%\"AG6#\"\"#" }}{PARA 4 "" 0 "" {TEXT 
-1 12 "   ---------" }{TEXT 256 24 "------------------------" }}{PARA 
4 "" 0 "" {TEXT 257 19 "             |     " }{XPPEDIT 18 0 "1/4;" "6#
*&\"\"\"F$\"\"%!\"\"" }{TEXT 258 10 "          " }{XPPEDIT 18 0 "3/4;
" "6#*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT 259 1 " " }}{PARA 4 "" 0 "" 
{TEXT -1 81 "a) B\372squese el mayor orden alcanzable por los miembros
 de esta familia, indicando" }}{PARA 4 "" 0 "" {TEXT -1 15 "los valore
s de " }{XPPEDIT 18 0 "C;" "6#%\"CG" }{TEXT -1 3 " , " }{XPPEDIT 18 0 
"A[1];" "6#&%\"AG6#\"\"\"" }{TEXT -1 4 " y  " }{XPPEDIT 18 0 "A[2];" "
6#&%\"AG6#\"\"#" }{TEXT -1 26 "  para los que se consigue" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 40 "(desde aqu\355 es \+
Ejercicio del cap\355tulo 8)" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 
4 "" 0 "" {TEXT -1 57 "b) Calc\372lese la funci\363n de estabilidad en
 el caso general" }}{PARA 4 "" 0 "" {TEXT -1 63 "c) B\372squese un eje
mplo de m\351todo A-estable dentro de la familia" }}{PARA 4 "" 0 "" 
{TEXT -1 24 "de m\351todos del problema." }}{PARA 4 "" 0 "" {TEXT -1 
63 "d) Compru\351bese si es o no es A-estable el m\351todo con valores
   " }{XPPEDIT 18 0 "A[1];" "6#&%\"AG6#\"\"\"" }{TEXT -1 10 "= 2/3  y \+
 " }{XPPEDIT 18 0 "A[2];" "6#&%\"AG6#\"\"#" }{TEXT -1 2 "=2" }}{PARA 
4 "" 0 "" {TEXT -1 20 "de los par\341metros ( " }{XPPEDIT 18 0 "C;" "6
#%\"CG" }{TEXT -1 70 "  no influye, claro) y dib\372jese la regi\363n \+
de estabilidad en este caso." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "interface(
labeling=false):" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 2 "a)" }{MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 37 "# desarrollo de l
a verdadera solucion" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "#y:
=unapply(y(x),x):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "#f:=un
apply(f(x,y),x,y):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "f0:=f
(x,y(x)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "f1:=subs(diff(
y(x),x)=f(x,y(x)),diff(f0,x)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 44 "f2:=subs(diff(y(x),x)=f(x,y(x)),diff(f1,x)):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 44 "f3:=subs(diff(y(x),x)=f(x,y(x)),diff(f2,x
)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "f0:=subs(\{y(x)=y0,x
=x0\},f0):f0:=expand(f0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
43 "f1:=subs(\{y(x)=y0,x=x0\},f1):f1:=expand(f1):" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 43 "f2:=subs(\{y(x)=y0,x=x0\},f2):f2:=expand(f2)
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "f3:=subs(\{y(x)=y0,x=x
0\},f3):f3:=expand(f3):" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 23 "#
 desarrollo del m\351todo" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
25 "k10:=unapply(f(x0,y0),h):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 59 "k20:=unapply(f(x0+C*h,y0+A[1]*h*kp10(h)+A[2]*h*kp20(h)),h):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "k100:=k10(0):k100:=expand(k1
00):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "k200:=k20(0):k200:=
expand(k200):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Fi0:=expan
d((1/4)*k100+(3/4)*k200):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
12 "k11:=D(k10):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "k21:=D(
k20):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "k110:=subs(\{kp10(
0)=k100,kp20(0)=k200\},k11(0)):k110:=expand(k110):" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 66 "k210:=subs(\{kp10(0)=k100,kp20(0)=k200\},k2
1(0)):k210:=expand(k210):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
35 "Fi1:=expand((1/4)*k110+(3/4)*k210):" }}}{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 98 "k120:
=subs(\{D(kp10)(0)=k110,D(kp20)(0)=k210,kp10(0)=k100,kp20(0)=k200\},k1
2(0)):k120:=expand(k120):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
98 "k220:=subs(\{D(kp10)(0)=k110,D(kp20)(0)=k210,kp10(0)=k100,kp20(0)=
k200\},k22(0)):k220:=expand(k220):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "Fi2:=expand((1/4)*k120+(3/4)*k220):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 12 "k13:=D(k12):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 12 "k23:=D(k22):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 140 "k130:=subs(\{(D@@2)(kp10)(0)=k120,(D@@2)(kp20)(0)=k220,D(kp10
)(0)=k110,D(kp20)(0)=k210,kp10(0)=k100,kp20(0)=k200\},k13(0)):k130:=ex
pand(k130):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "k230:=subs(
\{(D@@2)(kp10)(0)=k120,(D@@2)(kp20)(0)=k220,D(kp10)(0)=k110,D(kp20)(0)
=k210,kp10(0)=k100,kp20(0)=k200\},k23(0)):k230:=expand(k230):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Fi3:=expand((1/4)*k130+(3/4)
*k230):" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 24 "# consistencia y \+
orden 1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "cond1a:=coeff(Fi
0,f(x0,y0))=coeff(f0,f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'
cond1aG/\"\"\"F&" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 29 "# condic
iones para el orden 2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "co
nd2a:=coeff(Fi1,D[1](f)(x0,y0))=coeff((1/2)*f1,D[1](f)(x0,y0));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond2aG/,$*(\"\"$\"\"\"\"\"%!\"\"%
\"CGF)F)#F)\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "cond2b:
=coeff(coeff(Fi1,D[2](f)(x0,y0)),f(x0,y0))=coeff(coeff((1/2)*f1,D[2](f
)(x0,y0)),f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond2bG/,&*
&#\"\"$\"\"%\"\"\"&%\"AG6#F+F+F+*&F(F+&F-6#\"\"#F+F+#F+F2" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "orden2:=solve(\{cond2a,cond2b\});" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'orden2G<%/&%\"AG6#\"\"#,&&F(6#\"
\"\"!\"\"#F*\"\"$F./%\"CGF0/F,F," }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 
0 21 29 "# condiciones para el orden 3" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 69 "cond3a:=coeff(Fi2,D[1,1](f)(x0,y0))=coeff((1/3)*f2,D[
1,1](f)(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3aG/,$*(\"
\"$\"\"\"\"\"%!\"\"%\"CG\"\"#F)#F)F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 101 "cond3b:=coeff(coeff(Fi2,D[1,2](f)(x0,y0)),f(x0,y0))=
coeff(coeff((1/3)*f2,D[1,2](f)(x0,y0)),f(x0,y0));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'cond3bG/,&*&#\"\"$\"\"#\"\"\"*&%\"CGF+&%\"AG6#F+F+F+
F+*&F(F+*&F-F+&F/6#F*F+F+F+#F*F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 105 "cond3c:=coeff(coeff(Fi2,D[2,2](f)(x0,y0)),f(x0,y0)^2
)=coeff(coeff((1/3)*f2,D[2,2](f)(x0,y0)),f(x0,y0)^2);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%'cond3cG/,(*&#\"\"$\"\"%\"\"\"*$)&%\"AG6#F+\"\"#
F+F+F+*&#F)F1F+*&F.F+&F/6#F1F+F+F+*&F(F+*$)F5F1F+F+F+#F+F)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "cond3d:=coeff(coeff(Fi2,D[2](f)(x0
,y0)),D[1](f)(x0,y0))=coeff(coeff((1/3)*f2,D[2](f)(x0,y0)),D[1](f)(x0,
y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3dG/,$*&#\"\"$\"\"#\"
\"\"*&%\"CGF+&%\"AG6#F*F+F+F+#F+F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 101 "cond3e:=coeff(coeff(Fi2,D[2](f)(x0,y0)^2),f(x0,y0))=
coeff(coeff((1/3)*f2,D[2](f)(x0,y0)^2),f(x0,y0));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'cond3eG/,&*&#\"\"$\"\"#\"\"\"*&&%\"AG6#F+F+&F.6#F*F+
F+F+*&F(F+*$)F0F*F+F+F+#F+F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 66 "orden3:=solve(\{cond2a,cond2b,cond3a,cond3b,cond3c,cond3d,cond3e
\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'orden3G<%/%\"CG#\"\"#\"\"$/
&%\"AG6#\"\"\"#F/F*/&F-6#F)F0" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 
21 36 "se obtiene orden 3 para los valores " }{XPPEDIT 18 0 "C;" "6#%
\"CG" }{MPLTEXT 0 21 10 " = 2/3 ,  " }{XPPEDIT 18 0 "A[1];" "6#&%\"AG6
#\"\"\"" }{MPLTEXT 0 21 11 " = 1/3  y  " }{XPPEDIT 18 0 "A[2];" "6#&%
\"AG6#\"\"#" }{MPLTEXT 0 21 9 " = 1/3   " }}}{EXCHG {PARA 0 "" 0 "" 
{MPLTEXT 0 21 29 "# condiciones para el orden 4" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 73 "cond4a:=coeff(Fi3,D[1,1,1](f)(x0,y0))=coeff((1
/4)*f3,D[1,1,1](f)(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond
4aG/,$*(\"\"$\"\"\"\"\"%!\"\"%\"CGF(F)#F)F*" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 105 "cond4b:=coeff(coeff(Fi3,D[1,1,2](f)(x0,y0)),f(x0,
y0))=coeff(coeff((1/4)*f3,D[1,1,2](f)(x0,y0)),f(x0,y0));" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%'cond4bG/,&*&#\"\"*\"\"%\"\"\"*&)%\"CG\"\"#F+&
%\"AG6#F+F+F+F+*&F(F+*&F-F+&F16#F/F+F+F+#\"\"$F*" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 109 "cond4c:=coeff(coeff(Fi3,D[1,2,2](f)(x0,y0)),
f(x0,y0)^2)=coeff(coeff((1/4)*f3,D[1,2,2](f)(x0,y0)),f(x0,y0)^2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond4cG/,(*&#\"\"*\"\"#\"\"\"*(%\"C
GF+&%\"AG6#F+F+&F/6#F*F+F+F+*&#F)\"\"%F+*&F-F+)F.F*F+F+F+*&F4F+*&F-F+)
F1F*F+F+F+#\"\"$F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "cond
4d:=coeff(coeff(Fi3,D[1,2](f)(x0,y0)),D[1](f)(x0,y0))=coeff(coeff((1/4
)*f3,D[1,2](f)(x0,y0)),D[1](f)(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%'cond4dG/,$*&#\"\"*\"\"#\"\"\"*&)%\"CGF*F+&%\"AG6#F*F+F+F+#\"
\"$\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "cond4e:=coeff(
coeff(coeff(Fi3,D[1,2](f)(x0,y0)),D[2](f)(x0,y0)),f(x0,y0))=coeff(coef
f(coeff((1/4)*f3,D[1,2](f)(x0,y0)),D[2](f)(x0,y0)),f(x0,y0));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond4eG/,&**\"\"*\"\"\"%\"CGF)&%\"A
G6#F)F)&F,6#\"\"#F)F)*(F(F)F*F))F.F0F)F)#\"\"&\"\"%" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 109 "cond4f:=coeff(coeff(Fi3,D[2,2,2](f)(x0,y
0)),f(x0,y0)^3)=coeff(coeff((1/4)*f3,D[2,2,2](f)(x0,y0)),f(x0,y0)^3);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond4fG/,**&#\"\"$\"\"%\"\"\"*$
)&%\"AG6#\"\"#F)F+F+F+*&F(F+*$)&F/6#F+F)F+F+F+*&#\"\"*F*F+*&)F5F1F+F.F
+F+F+*&F8F+*&F5F+)F.F1F+F+F+#F+F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 145 "cond4g:=coeff(coeff(coeff(Fi3,D[2,2](f)(x0,y0)),D[1]
(f)(x0,y0)),f(x0,y0))=coeff(coeff(coeff((1/4)*f3,D[2,2](f)(x0,y0)),D[1
](f)(x0,y0)),f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond4gG/
,&*&#\"\"*\"\"#\"\"\"*(%\"CGF+&%\"AG6#F+F+&F/6#F*F+F+F+*&F(F+*&F-F+)F1
F*F+F+F+#\"\"$\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "con
d4h:=coeff(coeff(coeff(Fi3,D[2,2](f)(x0,y0)),D[2](f)(x0,y0)),f(x0,y0)^
2)=coeff(coeff(coeff((1/4)*f3,D[2,2](f)(x0,y0)),D[2](f)(x0,y0)),f(x0,y
0)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond4hG/,(*&#\"#F\"\"%\"
\"\"*&)&%\"AG6#F+\"\"#F+&F/6#F1F+F+F+*&#F)F1F+*&F.F+)F2F1F+F+F+*&F(F+*
$)F2\"\"$F+F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "cond4
i:=coeff(coeff(Fi3,D[1,1](f)(x0,y0)),D[2](f)(x0,y0))=coeff(coeff((1/4)
*f3,D[1,1](f)(x0,y0)),D[2](f)(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%'cond4iG/,$*&#\"\"*\"\"%\"\"\"*&)%\"CG\"\"#F+&%\"AG6#F/F+F+F+#
F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "cond4j:=coeff(coef
f(Fi3,D[1](f)(x0,y0)),D[2](f)(x0,y0)^2)=coeff(coeff((1/4)*f3,D[1](f)(x
0,y0)),D[2](f)(x0,y0)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond4j
G/,$*&#\"\"*\"\"#\"\"\"*&%\"CGF+)&%\"AG6#F*F*F+F+F+#F+\"\"%" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "cond4k:=coeff(coeff(Fi3,D[2
](f)(x0,y0)^3),f(x0,y0))=coeff(coeff((1/4)*f3,D[2](f)(x0,y0)^3),f(x0,y
0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond4kG/,&*&#\"\"*\"\"#\"\"
\"*$)&%\"AG6#F*\"\"$F+F+F+*&F(F+*&&F/6#F+F+)F.F*F+F+F+#F+\"\"%" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "subs(orden3,\{cond4a,cond4b,
cond4c,cond4d,cond4e,cond4f,cond4g,cond4h,cond4i,cond4j,cond4k\});" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#<'/\"\"\"F%/#\"\"#\"\"*#F%\"\"%/#F(\"
\"$#F.F+/#F+F.#\"\"&F+/#F%F.F*" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 
21 81 "y no se puede alcanzar orden 4 , lo que por otra parte era obvi
o ya que el m\351todo" }}{PARA 0 "" 0 "" {MPLTEXT 0 21 62 "de GAUSS de
 2 etapas no est\341 entre los de la familia propuesta" }}}{EXCHG 
{PARA 4 "" 0 "" {TEXT -1 68 "Se obtiene orden m\341ximo  3  de una sol
a manera, para los valores    " }{XPPEDIT 18 0 "C;" "6#%\"CG" }{TEXT 
-1 9 "= 2/3 ,  " }{XPPEDIT 18 0 "A[1];" "6#&%\"AG6#\"\"\"" }{TEXT -1 
11 " = 1/3  y  " }{XPPEDIT 18 0 "A[2];" "6#&%\"AG6#\"\"#" }{TEXT -1 
14 " = 1/3 , o sea" }}{PARA 4 "" 0 "" {TEXT -1 14 "para el m\351todo" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 268 22 "    0 \+
    |          0" }}{PARA 4 "" 0 "" {TEXT 269 4 "    " }{XPPEDIT 18 0 
"2/3;" "6#*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT 270 14 "     |        " }
{XPPEDIT 18 0 "1/3;" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT 274 10 "         \+
 " }{XPPEDIT 18 0 "1/3;" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT 275 10 "     \+
     " }}{PARA 4 "" 0 "" {TEXT -1 12 "   ---------" }{TEXT 266 44 "---
-----------------------------------------" }}{PARA 4 "" 0 "" {TEXT 
267 21 "            |        " }{XPPEDIT 18 0 "1/4;" "6#*&\"\"\"F$\"\"
%!\"\"" }{TEXT 272 10 "          " }{XPPEDIT 18 0 "3/4;" "6#*&\"\"$\"
\"\"\"\"%!\"\"" }{TEXT 273 1 " " }}{PARA 4 "" 0 "" {TEXT -1 61 "que su
ele recibir el nombre espec\355fico de RADAU I de 2 etapas" }}}{EXCHG 
{PARA 4 "" 0 "" {TEXT -1 2 "b)" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }{MPLTEXT 0 21 26 "# funcion 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 25 "
ka1:=unapply(f(x0,y0),h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ka1Gf*
6#%\"hG6\"6$%)operatorG%&arrowGF(*&%'lambdaG\"\"\"%#y0GF.F(F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "ka2:=unapply(f(x0+C*h,y0+A[1
]*h*kp1(h)+A[2]*h*kp2(h)),h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ka
2Gf*6#%\"hG6\"6$%)operatorG%&arrowGF(*&%'lambdaG\"\"\",(%#y0GF.*(&%\"A
G6#F.F.9$F.-%$kp1G6#F5F.F.*(&F36#\"\"#F.F5F.-%$kp2GF8F.F.F.F(F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "test:=solve(\{k1=subs(\{kp1(
h)=k1,kp2(h)=k2\},ka1(h)),k2=subs(\{kp1(h)=k1,kp2(h)=k2\},ka2(h))\},\{
k1,k2\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%testG<$/%#k1G*&%'lambd
aG\"\"\"%#y0GF*/%#k2G,$**F)F*F+F*,&F*F**(F)F*&%\"AG6#F*F*%\"hGF*F*F*,&
F*!\"\"*(F)F*&F36#\"\"#F*F5F*F*F7F7" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 36 "k1:=subs(test,k1);k2:=subs(test,k2);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#k1G*&%'lambdaG\"\"\"%#y0GF'" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#k2G,$**%'lambdaG\"\"\"%#y0GF(,&F(F(*(F'F(&%\"AG6#F
(F(%\"hGF(F(F(,&F(!\"\"*(F'F(&F-6#\"\"#F(F/F(F(F1F1" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 37 "y1:=expand(y0+h*((1/4)*k1+(3/4)*k2));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G,*%#y0G\"\"\"**\"\"%!\"\"%\"hGF'
%'lambdaGF'F&F'F'*&#\"\"$F)F'**F+F'F,F'F&F',&F'F**(F,F'&%\"AG6#\"\"#F'
F+F'F'F*F'F**&#F/F)F'*,F,F6F&F'F1F*&F46#F'F'F+F6F'F*" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 36 "y1:=collect(subs(lambda=z/h,y1),y0);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G*&,*\"\"\"F'*&\"\"%!\"\"%\"zGF'F
'*&#\"\"$F)F'*&F+F',&F'F**&F+F'&%\"AG6#\"\"#F'F'F*F'F**&#F.F)F'*(F+F5F
0F*&F36#F'F'F'F*F'%#y0GF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
57 "r:=unapply(collect(simplify(y1/y0),z),z):r_general:=r(z);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*r_generalG,$*&#\"\"\"\"\"%F(*&,(F)F
(*&,&&%\"AG6#\"\"#!\"\"*&\"\"$F(&F/6#F(F(F(F()%\"zGF1F(F(*&,&*&F)F(F.F
(F2F)F(F(F8F(F(F(,&F(F2*&F8F(F.F(F(F2F(F2" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 43 "r_nulo:=simplify(subs(\{A[2]=3*A[1]\},r(z)));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'r_nuloG*&,(\"\"\"!\"\"*(\"\"$F'%\"z
GF'&%\"AG6#F'F'F'F+F(F',&F'F(*(F*F'F+F'F,F'F'F(" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 2 "c)" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 17 "# A-esta
bilidad  " }}}{EXCHG {PARA 4 "" 0 "" {TEXT 276 40 "Estos m\351todos (i
ncluyendo el ya citado  " }{TEXT -1 20 "RADAU I de 2 etapas)" }{TEXT 
277 30 "  nunca pueden ser A-estables," }}{PARA 4 "" 0 "" {TEXT 278 
84 "ya que el numerador de  r(z)  es de grado 2 mientras que el denomi
nador lo es de  1 " }}{PARA 4 "" 0 "" {TEXT 279 25 "La excepci\363n po
dr\355a ser  " }{XPPEDIT 18 0 "A[2] = 3*A[1];" "6#/&%\"AG6#\"\"#*&\"\"
$\"\"\"&F%6#F*F*" }{TEXT 280 53 "  En este \372ltimo caso, la funci
\363n de estabilidad vale" }}{PARA 4 "" 0 "" {TEXT 281 8 "   r(z)=" }
{XPPEDIT 18 0 "(1+(1-3*A[1])*z)/(1-3*A[1]*z);" "6#*&,&\"\"\"F%*&,&F%F%
*&\"\"$F%&%\"AG6#F%F%!\"\"F%%\"zGF%F%F%,&F%F%*(F)F%&F+6#F%F%F.F%F-F-" 
}}{PARA 4 "" 0 "" {TEXT 282 43 "y es en principio de consecuencias dud
osas." }{TEXT -1 0 "" }}}{EXCHG {PARA 4 "" 0 "" {TEXT 285 21 "Si hacem
os entonces  " }{XPPEDIT 18 0 "A[1];" "6#&%\"AG6#\"\"\"" }{TEXT 286 
44 " =1/3  obtenemos como funci\363n de estabilidad" }}{PARA 4 "" 0 "
" {TEXT 288 8 "   r(z)=" }{XPPEDIT 18 0 "1/(1-z);" "6#*&\"\"\"F$,&F$F$
%\"zG!\"\"F'" }{TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 287 66 " y ya sab
emos que dicho m\351todo es A-estable. Se trata del m\351todo  " }}
{PARA 4 "" 0 "" {TEXT 294 15 "      0   |   0" }}{PARA 4 "" 0 "" 
{TEXT 291 6 "      " }{XPPEDIT 18 0 "C;" "6#%\"CG" }{TEXT 295 7 "   | \+
  " }{XPPEDIT 18 0 "1/3;" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT 296 8 "    1
   " }}{PARA 4 "" 0 "" {TEXT -1 12 "   ---------" }{TEXT 289 10 "-----
-----" }}{PARA 4 "" 0 "" {TEXT 290 14 "          |   " }{XPPEDIT 18 0 
"1/4;" "6#*&\"\"\"F$\"\"%!\"\"" }{TEXT 292 4 "    " }{XPPEDIT 18 0 "3/
4;" "6#*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT 293 1 " " }{TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "r_ejemplo1:=simplify(subs(\{
A[1]=1/3\},r_nulo));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+r_ejemplo1G
,$*&\"\"\"F',&F'!\"\"%\"zGF'F)F)" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 
2 "d)" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 6 "Para  \+
" }{XPPEDIT 18 0 "A[1];" "6#&%\"AG6#\"\"\"" }{TEXT -1 10 "= 2/3  y  " 
}{XPPEDIT 18 0 "A[2];" "6#&%\"AG6#\"\"#" }{TEXT -1 40 "= 2 , que entra
 dentro de la situaci\363n  " }{TEXT 297 1 " " }{XPPEDIT 18 0 "A[2] = \+
3*A[1];" "6#/&%\"AG6#\"\"#*&\"\"$\"\"\"&F%6#F*F*" }{TEXT -1 6 ",  la \+
" }{TEXT 298 25 "funci\363n de estabilidad es" }}{PARA 4 "" 0 "" 
{TEXT 299 8 "   r(z)=" }{XPPEDIT 18 0 "(z-1)/(2*z-1);" "6#*&,&%\"zG\"
\"\"F&!\"\"F&,&*&\"\"#F&F%F&F&F&F'F'" }{TEXT -1 0 "" }}{PARA 4 "" 0 "
" {TEXT -1 14 "              " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 53 "r_ejemplo2:=simplify(subs(\{A[1]=2/3,A[1]=2\}
,r_nulo));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+r_ejemplo2G*&,&\"\"\"
!\"\"%\"zGF'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 4 "" 0 "" {TEXT 300 21 "Pongamos   z=x+i y . " }{TEXT -1 
4 "Como" }}{PARA 4 "" 0 "" {TEXT -1 39 "     | 2 z - 1 |^2 = (2 x -1)^
2 + 4 y^2" }}{PARA 4 "" 0 "" {TEXT -1 4 "y   " }}{PARA 4 "" 0 "" 
{TEXT -1 33 "     | z - 1 |^2 = (x -1)^2 + y^2" }}{PARA 4 "" 0 "" 
{TEXT -1 78 "la frontera  | r(z) | = 1  o mejor  | r(z) |^2 = 1  del d
ominio de estabilidad" }}{PARA 4 "" 0 "" {TEXT -1 23 "es la curva de e
cuaci\363n" }}{PARA 4 "" 0 "" {TEXT -1 39 "    (2 x -1)^2 + 4 y^2 = (x
 -1)^2 + y^2" }}{PARA 4 "" 0 "" {TEXT -1 39 "que, tras operar, es la c
ircunferencia " }}{PARA 4 "" 0 "" {TEXT -1 9 "    (x - " }{XPPEDIT 18 
0 "1/3;" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT -1 13 " )^2 + y^2 = " }
{XPPEDIT 18 0 "1/9;" "6#*&\"\"\"F$\"\"*!\"\"" }{TEXT -1 2 "  " }}
{PARA 4 "" 0 "" {TEXT -1 10 "de centro " }{XPPEDIT 18 0 "1/3;" "6#*&\"
\"\"F$\"\"$!\"\"" }{TEXT -1 11 "  y radio  " }{XPPEDIT 18 0 "1/3;" "6#
*&\"\"\"F$\"\"$!\"\"" }{TEXT -1 56 " , que pasa por  ( 0 , 0 ) . El va
lor  z= - 1  queda en " }}{PARA 4 "" 0 "" {TEXT -1 79 "el exterior de \+
la circunferencia, y, para \351l, r(-1)= 2/3 , loego  -1  pertenece" }
}{PARA 4 "" 0 "" {TEXT -1 53 "al dominio de estabilidad.  El dominio d
e estabilidad" }}{PARA 4 "" 0 "" {TEXT -1 63 "es el exterior de esa ci
rcunferencia, y el m\351todo es  A-estable" }}}}{MARK "0 14 0" 39 }
{VIEWOPTS 1 1 0 3 4 1802 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }