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{SECT 0 {EXCHG {PARA 4 "" 0 "" {TEXT 266 14 "Ejercicios de " }}{PARA 
4 "" 0 "" {TEXT 267 55 "(7) M\351todos de RUNGE-KUTTA impl\355citos y \+
semi-impl\355citos" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" 
{TEXT 265 55 "Ejercicio 07-05  (del EXAMEN EXTRAORDINARIO de 14JUL04)
" }}{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 272 44 "      1    |         1          0      \+
  0  " }}{PARA 4 "" 0 "" {TEXT 256 5 "     " }{XPPEDIT 18 0 "1/2;" "6#
*&\"\"\"F$\"\"#!\"\"" }{TEXT 271 10 "    |     " }{XPPEDIT 18 0 "-1/2;
" "6#,$*&\"\"\"F%\"\"#!\"\"F'" }{TEXT 269 20 "          1        0" }}
{PARA 4 "" 0 "" {TEXT 273 42 "      0    |         a        - 3       \+
 1" }}{PARA 4 "" 0 "" {TEXT -1 12 "   ---------" }{TEXT 257 26 "------
--------------------" }}{PARA 4 "" 0 "" {TEXT 258 21 "           |    \+
     " }{XPPEDIT 18 0 "1/6;" "6#*&\"\"\"F$\"\"'!\"\"" }{TEXT 262 10 " \+
         " }{XPPEDIT 18 0 "2/3;" "6#*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT 
263 7 "       " }{XPPEDIT 18 0 "1/6;" "6#*&\"\"\"F$\"\"'!\"\"" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 75 "y teniend
o en cuenta que para tales m\351todos de  3  etapas  se puede obtener
" }}{PARA 4 "" 0 "" {TEXT -1 72 "hasta orden  6 , d\355gase cu\341l es
 el orden m\341ximo que se obtiene realmente" }}{PARA 4 "" 0 "" {TEXT 
-1 79 "y cu\341les son los valores del par\341metro  a  para los que s
e alcanza dicho orden." }}{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 "Calc\372l
ese la funci\363n de estabilidad para  los m\351todos de la familia " 
}}{PARA 4 "" 0 "" {TEXT -1 35 "y para los m\351todos de orden m\341xim
o." }}}{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 59 "k2_:=unapply(f(x0+(1/2)*h,y0-(1/2)*
h*kp1_(h)+h*kp2_(h)),h):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 
"k3_:=unapply(f(x0,y0+a*h*kp1_(h)-3*h*kp2_(h)+h*kp3_(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 13 "k3_0:=k3_(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 12 "k31:=
D(k3_):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "k110:=subs(\{kp1
_(0)=k1_0,kp2_(0)=k2_0,kp3_(0)=k3_0\},k11(0)):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 60 "k210:=subs(\{kp1_(0)=k1_0,kp2_(0)=k2_0,kp3_(0)
=k3_0\},k21(0)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "k310:=s
ubs(\{kp1_(0)=k1_0,kp2_(0)=k2_0,kp3_(0)=k3_0\},k31(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 12 "k32:=D(k31):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 108 "k120:=subs(\{D(kp1_)(0)=k110,D(kp2_)(0)=k210,D(kp3_)(0)=k310,kp
1_(0)=k1_0,kp2_(0)=k2_0,kp3_(0)=k3_0\},k12(0)):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 108 "k220:=subs(\{D(kp1_)(0)=k110,D(kp2_)(0)=k210,
D(kp3_)(0)=k310,kp1_(0)=k1_0,kp2_(0)=k2_0,kp3_(0)=k3_0\},k22(0)):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "k320:=subs(\{D(kp1_)(0)=k11
0,D(kp2_)(0)=k210,D(kp3_)(0)=k310,kp1_(0)=k1_0,kp2_(0)=k2_0,kp3_(0)=k3
_0\},k32(0)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 27 "condiciones par
a el orden 1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Fi_0:=expan
d((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(x0,y0));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%'cond1aG/\"\"\"F&" }}}{EXCHG {PARA 256 "" 0 "" 
{TEXT -1 33 "El orden es siempre de 1 al menos" }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 260 27 "condiciones para el orden 2" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Fi10:=expand((1/6)*k110+(2/3
)*k210+(1/6)*k310);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Fi10G,(*&#\"
\"\"\"\"#F(--&%\"DG6#F(6#%\"fG6$%#x0G%#y0GF(F(*&#F(\"\"'F(*&--&F-6#F)F
/F1F(-F0F1F(F(F(*&F5F(*(F8F(%\"aGF(F<F(F(F(" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 66 "cond2a:=(1/2)*coeff(Y2,D[1](f)(x0,y0))=coeff(Fi10,
D[1](f)(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond2aG/#\"\"\"
\"\"#F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "cond2b:=(1/2)*co
eff(coeff(Y2,D[2](f)(x0,y0)),f(x0,y0))=coeff(coeff(Fi10,D[2](f)(x0,y0)
),f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond2bG/#\"\"\"\"\"
#,&#F'\"\"'F'*&F+!\"\"%\"aGF'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 33 "solucion:=solve(\{cond2a,cond2b\});" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%)solucionG<#/%\"aG\"\"#" }}}{EXCHG {PARA 256 "" 0 "" 
{TEXT -1 55 "El valor  a  = 2   es el \372nico que proporciona orden 2
 " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 261 28 "condicio
nes para el orden 3 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Fi2
0:=expand((1/6)*k120+(2/3)*k220+(1/6)*k320);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%%Fi20G,4*&#\"\"\"\"\"$F(--&%\"DG6$F(F(6#%\"fG6$%#x0G%
#y0GF(F(*&#\"\"#F)F(*&--&F-6$F(F6F/F1F(-F0F1F(F(F(*&--&F-6$F6F6F/F1F()
F<F6F(F(*&#F(\"\"'F(*&--&F-6#F6F/F1F(--&F-6#F(F/F1F(F(!\"\"*&#\"\"&FEF
(*&)FGF6F(F<F(F(FO*&#F(FEF(*(F>F()%\"aGF6F(FBF(F(F(*&#F6F)F(*(F>F(FYF(
FBF(F(FO*&F'F(*(FGF(FYF(FKF(F(F(*&F5F(*(FTF(FYF(F<F(F(F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "cond3a:=(1/6)*coeff(Y3,D[1,1](f)(x
0,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(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)):cond3b:=subs(solucion,cond3b);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'cond3bG/#\"\"\"\"\"$F&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 142 "cond3c:=(1/6)*coeff(coeff(Y3,D[2,2](f)(x0,y0)),f(x0,
y0)^2)=(1/2)*coeff(coeff(Fi20,D[2,2](f)(x0,y0)),f(x0,y0)^2):cond3c:=su
bs(solucion,cond3c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3cG/#\"
\"\"\"\"'F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "cond3d:=(1/
6)*coeff(coeff(Y3,D[2](f)(x0,y0)),D[1](f)(x0,y0))=(1/2)*coeff(coeff(Fi
20,D[2](f)(x0,y0)),D[1](f)(x0,y0)):cond3d:=subs(solucion,cond3d);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3dG/#\"\"\"\"\"'#F'\"\"%" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "cond3e:=(1/6)*coeff(coeff(Y
3,D[2](f)(x0,y0)^2),f(x0,y0))=(1/2)*coeff(coeff(Fi20,D[2](f)(x0,y0)^2)
,f(x0,y0)):cond3e:=subs(solucion,cond3e);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'cond3eG/#\"\"\"\"\"'#F'\"\"%" }}}{EXCHG {PARA 256 "
" 0 "" {TEXT -1 29 "Nunca se obtiene el orden 3  " }{MPLTEXT 1 0 0 "" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 23 "funci\363n de estabilidad " }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 19 "f:=(x,y)->lambda*y;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF)*&%'lambd
aG\"\"\"9%F/F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "ka1:=
unapply(f(x0+h,y0+h*kp1(h)),h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$
ka1Gf*6#%\"hG6\"6$%)operatorG%&arrowGF(*&%'lambdaG\"\"\",&%#y0GF.*&9$F
.-%$kp1G6#F2F.F.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 
"test:=solve(\{k1=subs(kp1(h)=k1,ka1(h))\},k1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%testG<#/%#k1G,$*(%'lambdaG\"\"\"%#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,$*(%'lambdaG\"\"
\"%#y0GF(,&F(!\"\"*&F'F(%\"hGF(F(F+F+" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 63 "ka2:=simplify(unapply(f(x0+(1/2)*h,y0-(1/2)*h*k1+h*kp
2(h)),h));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ka2Gf*6#%\"hG6\"6$%)o
peratorG%&arrowGF(*&%'lambdaG\"\"\",(%#y0GF.*&#F.\"\"#F.**9$F.F-F.F0F.
,&F.!\"\"*&F-F.F5F.F.F7F.F.*&F5F.-%$kp2G6#F5F.F.F.F(F(F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "test:=solve(\{k2=subs(kp2(h)=k2,ka2
(h))\},k2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%testG<#/%#k2G,$*,\"
\"#!\"\"%'lambdaG\"\"\"%#y0GF-,&F*F+*(\"\"$F-F,F-%\"hGF-F-F-,&F-F+*&F,
F-F2F-F-!\"#F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "k2:=subs(
test,k2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k2G,$*,\"\"#!\"\"%'lam
bdaG\"\"\"%#y0GF*,&F'F(*(\"\"$F*F)F*%\"hGF*F*F*,&F*F(*&F)F*F/F*F*!\"#F
(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "ka3:=simplify(unapply(
f(x0,y0+a*h*k1-3*h*k2+h*kp3(h)),h));" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%$ka3Gf*6#%\"hG6\"6$%)operatorG%&arrowGF(*&%'lambdaG\"\"\",*%#y0GF.
*,%\"aGF.9$F.F-F.F0F.,&F.!\"\"*&F-F.F3F.F.F5F5*&#\"\"$\"\"#F.*,F3F.F-F
.F0F.,&F:F5*(F9F.F-F.F3F.F.F.F4!\"#F.F.*&F3F.-%$kp3G6#F3F.F.F.F(F(F(" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "test:=solve(\{k3=subs(kp3
(h)=k3,ka3(h))\},k3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%testG<#/%#
k3G,$*,\"\"#!\"\"%'lambdaG\"\"\"%#y0GF-,,F*F+*(\"#5F-F,F-%\"hGF-F-*(\"
#6F-)F,F*F-)F2F*F-F+**F*F-%\"aGF-F2F-F,F-F+**F*F-F8F-F6F-F5F-F-F-,&F-F
+*&F,F-F2F-F-!\"$F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "k3:=
subs(test,k3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k3G,$*,\"\"#!\"\"
%'lambdaG\"\"\"%#y0GF*,,F'F(*(\"#5F*F)F*%\"hGF*F**(\"#6F*)F)F'F*)F/F'F
*F(**F'F*%\"aGF*F/F*F)F*F(**F'F*F5F*F3F*F2F*F*F*,&F*F(*&F)F*F/F*F*!\"$
F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "y1:=expand(y0+h*((1/6
)*k1+(2/3)*k2+(1/6)*k3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G,4%
#y0G\"\"\"*,\"\"'!\"\"%\"hGF'%'lambdaGF'F&F',&F'F**&F,F'F+F'F'F*F**.\"
\"#F'\"\"$F*F+F'F,F'F&F'F-!\"#F'**F,F0F&F'F-F2F+F0F**,F)F*F+F'F,F'F&F'
F-!\"$F**.\"\"&F'F)F*F,F0F&F'F-F5F+F0F'*.\"#6F'\"#7F*F,F1F&F'F-F5F+F1F
**.F)F*F,F0F&F'F-F5%\"aGF'F+F0F**.F)F*F,F1F&F'F-F5F<F'F+F1F'" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "y1:=collect(subs(lambda=z/h,
y1),y0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G*&,4\"\"\"F'*(\"\"'!
\"\"%\"zGF',&F'F*F+F'F*F***\"\"#F'\"\"$F*F+F'F,!\"#F'*&F+F.F,F0F**(F)F
*F+F'F,!\"$F***\"\"&F'F)F*F+F.F,F3F'**\"#6F'\"#7F*F+F/F,F3F***F)F*F+F.
F,F3%\"aGF'F***F)F*F+F/F,F3F: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$%)operatorG%&arrowGF(,$*
&#\"\"\"\"#7F/*&,*F0!\"\"*&,&\"#8F3*&\"\"#F/%\"aGF/F/F/)9$\"\"$F/F/*&,
&F8F3*&F8F/F9F/F3F/)F;F8F/F/*&\"#CF/F;F/F/F/,&F/F3F;F/!\"$F/F/F(F(F(" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 64 "que es la funci\363n de estabi
lidad de la familia general propuesta" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 270 53 "funci\363n de estabilidad cuando  a = 2   (orden 2)     \+
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "r_2:=simplify(subs(\{a=
2\},r(z)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$r_2G,$*(\"\"%!\"\",*
F'\"\"\"*&\"\"$F*)%\"zGF,F*F**&\"\"#F*)F.F0F*F**&\"\")F*F.F*F(F*,&F*F(
F.F*!\"$F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0
 1 0" 9 }{VIEWOPTS 1 1 0 3 4 1802 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }