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{SECT 0 {EXCHG {PARA 4 "" 0 "" {TEXT 264 14 "Ejercicios de " }}{PARA 
4 "" 0 "" {TEXT 265 22 "(8) Estabilidad lineal" }}{PARA 4 "" 0 "" 
{TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 259 42 "Ejercicio
 08-19  (del EXAMEN ESP DE JUL05)" }}{PARA 4 "" 0 "" {TEXT -1 56 "\nD
\355gase si el m\351todo de RUNGE-KUTTA dado por el tablero " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 261 28 "      0    |  \+
   2         1" }}{PARA 4 "" 0 "" {TEXT 262 28 "      1    |     0    \+
     1" }}{PARA 4 "" 0 "" {TEXT -1 12 "   ---------" }{TEXT 256 17 "--
---------------" }}{PARA 4 "" 0 "" {TEXT 257 17 "            |    " }
{XPPEDIT 18 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 258 9 "         \+
" }{XPPEDIT 18 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 263 4 "    " }
}{PARA 4 "" 0 "" {TEXT -1 17 "es o no 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 0 21 24 "# funcion 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 76 "c[1]:=0:c[2]:=
1:a[1,1]:=2:a[1,2]:=1:a[2,1]:=0:a[2,2]:=1:b[1]:=1/2:b[2]:=1/2:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "ka1:=unapply(f(x0+c[1]*h,y0+
a[1,1]*h*kp1(h)+a[1,2]*h*kp2(h)),h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%$ka1Gf*6#%\"hG6\"6$%)operatorG%&arrowGF(*&%'lambdaG\"\"\",(%#y0GF.
*(\"\"#F.9$F.-%$kp1G6#F3F.F.*&F3F.-%$kp2GF6F.F.F.F(F(F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "ka2:=unapply(f(x0+c[2]*h,y0+a[2,1]*
h*kp1(h)+a[2,2]*h*kp2(h)),h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ka
2Gf*6#%\"hG6\"6$%)operatorG%&arrowGF(*&%'lambdaG\"\"\",&%#y0GF.*&9$F.-
%$kp2G6#F2F.F.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "t
est:=solve(\{k1=subs(\{kp1(h)=k1,kp2(h)=k2\},ka1(h)),k2=subs(\{kp1(h)=
k1,kp2(h)=k2\},ka2(h))\},\{k1,k2\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%%testG<$/%#k1G*(,(\"\"\"F**(\"\"$F*%'lambdaGF*%\"hGF*!\"\"*(\"\"#F
*)F-F1F*)F.F1F*F*F/F-F*%#y0GF*/%#k2G,$*(F-F*F4F*,&F/F**&F-F*F.F*F*F/F/
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "k1:=subs(test,k1);k2:=s
ubs(test,k2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k1G*(,(\"\"\"F'*(
\"\"$F'%'lambdaGF'%\"hGF'!\"\"*(\"\"#F')F*F.F')F+F.F'F'F,F*F'%#y0GF'" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k2G,$*(%'lambdaG\"\"\"%#y0GF(,&!
\"\"F(*&F'F(%\"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\"\"\"*,#F'\"\"#F'%\"hGF',(F'F'*(\"\"$F'%'lambdaGF'
F+F'!\"\"*(F*F')F/F*F')F+F*F'F'F0F/F'F&F'F'*&#F'F*F'**F+F'F/F'F&F',&F0
F'*&F/F'F+F'F'F0F'F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "y1:
=collect(subs(lambda=z/h,y1),y0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%#y1G*&,(\"\"\"F'*(#F'\"\"#F',(F'F'*&\"\"$F'%\"zGF'!\"\"*&F*F')F.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 41 "r:=unapply(collect(simplify(y1/y0),z),z);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rGf*6#%\"zG6\"6$%)operatorG%&arro
wGF(*&,&!\"\"\"\"\"9$F/F/,&F0\"\"#F/F.F.F(F(F(" }}}{EXCHG {PARA 4 "" 
0 "" {TEXT 260 37 "Supongamos que  z=x+i y  con  x<=0 . " }}{PARA 4 "
" 0 "" {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 53 "entonces tanto  2 x - 1  como  x - 1  son
 negativos y" }}{PARA 4 "" 0 "" {TEXT -1 22 "    2 x - 1 <=  x - 1 " }
}{PARA 4 "" 0 "" {TEXT -1 7 "o sea, " }}{PARA 4 "" 0 "" {TEXT -1 26 " \+
   (2 x -1)^2 >= (x -1)^2" }}{PARA 4 "" 0 "" {TEXT -1 2 "y " }}{PARA 
4 "" 0 "" {TEXT -1 40 "    (2 x -1)^2 + 4 y^2 >= (x -1)^2 + y^2" }}
{PARA 4 "" 0 "" {TEXT -1 5 "o sea" }}{PARA 4 "" 0 "" {TEXT -1 16 "   |
 r(z) | <= 1" }}{PARA 4 "" 0 "" {TEXT -1 58 "y el semiplano de parte r
eal negariva est\341 contenido en el" }}{PARA 4 "" 0 "" {TEXT -1 48 "d
ominio de estabilidad. El m\351todo es  A-estable." }}{PARA 4 "" 0 "" 
{TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 63 "De hecho, la frontera  | \+
r(z) | = 1  del dominio de estabilidad" }}{PARA 4 "" 0 "" {TEXT -1 23 
"es la curva de ecuaci\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, tra
s operar, es la circunferencia " }}{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 ) ,
 y el dominio de estabilidad" }}{PARA 4 "" 0 "" {TEXT -1 37 "es el ext
erior de esa circunferencia." }}}}{MARK "0 3 1" 15 }{VIEWOPTS 1 1 0 3 
4 1802 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }