{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 
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 24 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 275 "" 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 "Headi
ng 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 274 14 "Ejercicios de " }}{PARA 
4 "" 0 "" {TEXT 275 22 "(8) Estabilidad lineal" }}{PARA 4 "" 0 "" 
{TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 260 51 "Ejercicio 08-28   (del E
XAMEN ORDINARIO DE 24ENE11)" }}{PARA 4 "" 0 "" {TEXT -1 75 "Consid\351
rese la familia de m\351todos impl\355citos de RUNGE-KUTTA con dos eta
pas " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 263 5 " \+
    " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT 264 11 "     |     \+
" }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT 265 13 " - 1       1 " }
}{PARA 4 "" 0 "" {TEXT 266 20 "     1      |       " }{XPPEDIT 18 0 "b
eta;" "6#%%betaG" }{TEXT 271 13 "           0 " }}{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 16 "los valore
s de  " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 8 "  y de  " }
{XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT -1 26 "  para los que se cons
igue" }}{PARA 4 "" 0 "" {TEXT -1 40 "(hasta aqu\355 es Ejercicio del c
ap\355tulo 7)" }}{PARA 4 "" 0 "" {TEXT -1 67 "b) B\372squese un m\351t
odo de la familia del enunciado que sea A-estable" }}{PARA 4 "" 0 "" 
{TEXT -1 76 "c) Est\372diese la A-estabilidad de los m\351todos de ord
en m\341ximo obtenidos antes" }}}{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 0 21 37 "# desarr
ollo de la verdadera solucion" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 20 "#y:=unapply(y(x),x):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
24 "#f:=unapply(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:=sub
s(\{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:=su
bs(\{y(x)=y0,x=x0\},f3):f3:=expand(f3):" }}}{EXCHG {PARA 0 "" 0 "" 
{MPLTEXT 0 21 23 "# desarrollo del m\351todo" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 63 "k10:=unapply(f(x0+alpha*h,y0+(alpha-1)*h*kp10(h)
+h*kp20(h)),h):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "k20:=una
pply(f(x0+h,y0+beta*h*kp10(h)),h):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "k100:=k10(0):k100:=expand(k100):" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 32 "k200:=k20(0):k200:=expand(k200):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Fi0:=expand((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)=k
200\},k11(0)):k110:=expand(k110):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 66 "k210:=subs(\{kp10(0)=k100,kp20(0)=k200\},k21(0)):k210
:=expand(k210):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Fi1:=exp
and((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(kp1
0)(0)=k110,D(kp20)(0)=k210,kp10(0)=k100,kp20(0)=k200\},k12(0)):k120:=e
xpand(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)=k
210,kp10(0)=k100,kp20(0)=k200\},k13(0)):k130:=expand(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)=k1
00,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(Fi0,f(x0,y0))=coeff(f
0,f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond1aG/\"\"\"F&" }
}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 29 "# condiciones para el orden
 2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "cond2a:=coeff(Fi1,D[1
](f)(x0,y0))=coeff((1/2)*f1,D[1](f)(x0,y0));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'cond2aG/,&*&\"\"%!\"\"%&alphaG\"\"\"F+#\"\"$F(F+#F+
\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "cond2b:=coeff(coef
f(Fi1,D[2](f)(x0,y0)),f(x0,y0))=coeff(coeff((1/2)*f1,D[2](f)(x0,y0)),f
(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond2bG/,&*&\"\"%!\"\"
%&alphaG\"\"\"F+*(\"\"$F+F(F)%%betaGF+F+#F+\"\"#" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 32 "orden2:=solve(\{cond2a, cond2b\});" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%'orden2G<$/%&alphaG!\"\"/%%betaG\"\"\"" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "alpha:=subs(orden2,alpha);b
eta:=subs(orden2,beta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG!
\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG\"\"\"" }}}{EXCHG 
{PARA 0 "" 0 "" {MPLTEXT 0 21 29 "# condiciones para el orden 3" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "cond3a:=coeff(Fi2,D[1,1](f)(
x0,y0))=coeff((1/3)*f2,D[1,1](f)(x0,y0));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'cond3aG/\"\"\"#F&\"\"$" }}}{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/\"\"##F&\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 105 "cond3c:=coeff(coeff(Fi2,D[2,2](f)(x0,y0)),f(x0,y0)^2
)=coeff(coeff((1/3)*f2,D[2,2](f)(x0,y0)),f(x0,y0)^2);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%'cond3cG/\"\"\"#F&\"\"$" }}}{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/\"\"!#\"\"\"\"\"$" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "cond3e:=coeff(coeff(Fi2,D[2
](f)(x0,y0)^2),f(x0,y0))=coeff(coeff((1/3)*f2,D[2](f)(x0,y0)^2),f(x0,y
0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3eG/\"\"!#\"\"\"\"\"$" 
}}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 73 "No se puede alcanzar  orden 3 ;
 basta ver la primera de las condiciones, " }}{PARA 4 "" 0 "" {TEXT 
-1 46 "que es imposible. El m\341ximo orden es  2  con  " }{XPPEDIT 
18 0 "alpha = -1;" "6#/%&alphaG,$\"\"\"!\"\"" }{TEXT -1 5 "  y  " }
{XPPEDIT 18 0 "beta = 1;" "6#/%%betaG\"\"\"" }{TEXT -1 21 "           \+
          " }}{PARA 4 "" 0 "" {TEXT -1 66 "Se obtiene orden m\341ximo \+
 3  de una sola manera, para los valores  " }{XPPEDIT 18 0 "alpha = -1
;" "6#/%&alphaG,$\"\"\"!\"\"" }{TEXT -1 5 "  y  " }{XPPEDIT 18 0 "beta
 = 1;" "6#/%%betaG\"\"\"" }{TEXT -1 9 "  , o sea" }}{PARA 4 "" 0 "" 
{TEXT -1 14 "para el m\351todo" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 4 "" 0 "" {TEXT 269 34 "    -1     |        -2           1" }}
{PARA 4 "" 0 "" {TEXT 270 44 "     1     |         1           0      \+
    " }}{PARA 4 "" 0 "" {TEXT -1 12 "   ---------" }{TEXT 267 44 "----
----------------------------------------" }}{PARA 4 "" 0 "" {TEXT 268 
21 "            |        " }{XPPEDIT 18 0 "1/4;" "6#*&\"\"\"F$\"\"%!\"
\"" }{TEXT 272 10 "          " }{XPPEDIT 18 0 "3/4;" "6#*&\"\"$\"\"\"
\"\"%!\"\"" }{TEXT 273 1 " " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{MPLTEXT 0 21 30 "# funcion de estabilidad para " }{XPPEDIT 18 0 "alph
a = 1;" "6#/%&alphaG\"\"\"" }{MPLTEXT 0 21 3 " y " }{XPPEDIT 18 0 "bet
a = 1;" "6#/%%betaG\"\"\"" }{MPLTEXT 0 21 25 " (solo con orden  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 17 "alpha:=1;beta:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%&alphaG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG\"\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "k10:=
unapply(f(x0+alpha*h,y0+(alpha-1)*h*kp1(h)+h*kp2(h)),h);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%$k10Gf*6#%\"hG6\"6$%)operatorG%&arrowGF(*&%'la
mbdaG\"\"\",&%#y0GF.*&9$F.-%$kp2G6#F2F.F.F.F(F(F(" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 46 "k20:=unapply(f(x0+beta*h,y0+beta*h*kp1(h)),h
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$k20Gf*6#%\"hG6\"6$%)operatorG
%&arrowGF(*&%'lambdaG\"\"\",&%#y0GF.*&9$F.-%$kp1G6#F2F.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 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "ka1:=unapply(f(x0+alpha*h,y0
+(alpha-1)*h*kp1(h)+h*kp2(h)),h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%$ka1Gf*6#%\"hG6\"6$%)operatorG%&arrowGF(*&%'lambdaG\"\"\",&%#y0GF.*&9
$F.-%$kp2G6#F2F.F.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
46 "ka2:=unapply(f(x0+beta*h,y0+beta*h*kp1(h)),h);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%$ka2Gf*6#%\"hG6\"6$%)operatorG%&arrowGF(*&%'lambdaG
\"\"\",&%#y0GF.*&9$F.-%$kp1G6#F2F.F.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 99 "test:=solve(\{k1=subs(\{kp1(h)=k1,kp2(h)=k2\},ka
1(h)),k2=subs(\{kp1(h)=k1,kp2(h)=k2\},ka2(h))\},\{k1,k2\});" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%%testG<$/%#k2G,$*(%'lambdaG\"\"\"%#y0GF+,&
*&F*F+%\"hGF+F+F+!\"\"F0F0/%#k1GF(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 54 "k1:=subs(test,k1);k2:=subs(test,k2);simplify(k1+3*k2)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k1G,$*(%'lambdaG\"\"\"%#y0GF(,
&*&F'F(%\"hGF(F(F(!\"\"F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k2G,
$*(%'lambdaG\"\"\"%#y0GF(,&*&F'F(%\"hGF(F(F(!\"\"F-F-" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,$**\"\"%\"\"\"%'lambdaGF&%#y0GF&,&*&F'F&%\"hGF&F&
F&!\"\"F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "y1:=expand(y
0+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-" }}}
{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'%#y0GF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
60 "r:=unapply(collect(simplify(y1/y0),z),z):r_particular:=r(z);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-r_particularG,$*&\"\"\"F',&%\"zGF'F
'!\"\"F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{MPLTEXT 0 21 83 "# para estos valores  alpha = 1  y  beta = 1  el m
\351todo es evidentemente A-estable " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {MPLTEXT 0 
21 30 "# funcion de estabilidad para " }{XPPEDIT 18 0 "alpha = -1;" "6
#/%&alphaG,$\"\"\"!\"\"" }{MPLTEXT 0 21 3 " y " }{XPPEDIT 18 0 "beta =
 1;" "6#/%%betaG\"\"\"" }{MPLTEXT 0 21 23 " (orden  2  m\341ximo)    \+
" }}}{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 18 "alpha:=-1;beta:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%&alphaG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG\"\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "k10:=
unapply(f(x0+alpha*h,y0+(alpha-1)*h*kp1(h)+h*kp2(h)),h);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%$k10Gf*6#%\"hG6\"6$%)operatorG%&arrowGF(*&%'la
mbdaG\"\"\",(%#y0GF.*(\"\"#F.9$F.-%$kp1G6#F3F.!\"\"*&F3F.-%$kp2GF6F.F.
F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "k20:=unapply(f(
x0+beta*h,y0+beta*h*kp1(h)),h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$
k20Gf*6#%\"hG6\"6$%)operatorG%&arrowGF(*&%'lambdaG\"\"\",&%#y0GF.*&9$F
.-%$kp1G6#F2F.F.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "
ka1:=unapply(f(x0+alpha*h,y0+(alpha-1)*h*kp1(h)+h*kp2(h)),h);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ka1Gf*6#%\"hG6\"6$%)operatorG%&arro
wGF(*&%'lambdaG\"\"\",(%#y0GF.*(\"\"#F.9$F.-%$kp1G6#F3F.!\"\"*&F3F.-%$
kp2GF6F.F.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "ka2:=
unapply(f(x0+beta*h,y0+beta*h*kp1(h)),h);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$ka2Gf*6#%\"hG6\"6$%)operatorG%&arrowGF(*&%'lambdaG\"
\"\",&%#y0GF.*&9$F.-%$kp1G6#F2F.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<$/%#k1G,$**%'lambdaG\"\"\"%#y0GF+,&F+F+
*&F*F+%\"hGF+F+F+,(F+!\"\"*(\"\"#F+F*F+F/F+F1*&)F*F3F+)F/F3F+F+F1F1/%#
k2G,$**F*F+F,F+,&F+F+*(\"\"$F+F*F+F/F+F+F+F0F1F1" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 54 "k1:=subs(test,k1);k2:=subs(test,k2);simplify(
k1+3*k2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k1G,$**%'lambdaG\"\"\"
%#y0GF(,&F(F(*&F'F(%\"hGF(F(F(,(F(!\"\"*(\"\"#F(F'F(F,F(F.*&)F'F0F()F,
F0F(F(F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k2G,$**%'lambdaG\"\"
\"%#y0GF(,&F(F(*(\"\"$F(F'F(%\"hGF(F(F(,(F(!\"\"*(\"\"#F(F'F(F-F(F/*&)
F'F1F()F-F1F(F(F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*,\"\"#\"\"\"
%'lambdaGF&%#y0GF&,&F%F&*(\"\"&F&F'F&%\"hGF&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 
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'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'F+*$)F)F-F'F'F+F+**\"\"&F'F-F+F)F-F*F+F+F'
%#y0GF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "r:=unapply(colle
ct(simplify(y1/y0),z),z):r_particular:=r(z);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%-r_particularG,$*(\"\"#!\"\",(F'\"\"\"*&\"\"'F*%\"zGF
*F**&\"\"$F*)F-F'F*F*F*,(F*F(*&F'F*F-F*F(*$F0F*F*F(F(" }}}{EXCHG 
{PARA 0 "" 0 "" {MPLTEXT 0 21 73 "# para estos valores  alpha = -1  y \+
 beta = 1  el m\351todo no es A-estable," }}{PARA 0 "" 0 "" {MPLTEXT 
0 21 49 "ya que  r(z) tiende a  3/2  cuando  z  tiende a  " }{XPPEDIT 
18 0 "infinity;" "6#%)infinityG" }{MPLTEXT 0 21 3 "   " }}}}{MARK "0 1
2 0" 39 }{VIEWOPTS 1 1 0 3 4 1802 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }