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{SECT 0 {EXCHG {PARA 4 "" 0 "" {TEXT 257 14 "Ejercicios de " }}{PARA 
4 "" 0 "" {TEXT 258 46 "(5) Sistemas aut\363nomos: m\351todos de RUNGE
-KUTTA" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 256 
15 "Ejercicio 05-04" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "
" {TEXT -1 40 "Para el problema tipo aut\363nomo de Cauchy" }}{PARA 4 
"" 0 "" {TEXT -1 14 "     y'_1 =   " }{XPPEDIT 18 0 "(-y_1^2+y_2^4)/y_
2^2" "6#*&,&*$%$y_1G\"\"#!\"\"*$%$y_2G\"\"%\"\"\"F,*$F*F'F(" }{TEXT 
-1 4 "    " }}{PARA 4 "" 0 "" {TEXT -1 15 "     y'_2 =    " }{XPPEDIT 
18 0 "-y_1/y_2" "6#,$*&%$y_1G\"\"\"%$y_2G!\"\"F(" }{TEXT -1 6 "      \+
" }}{PARA 4 "" 0 "" {TEXT -1 28 "     y_1(0) = 0 , y_2(0) = 1" }}
{PARA 4 "" 0 "" {TEXT -1 88 "(de soluci\363n exacta   y_1(x) = sen(x) \+
cos(x) , y_2(x) = cos(x) ) util\355cese el m\351todo de " }}{PARA 4 "
" 0 "" {TEXT -1 23 "RUNGE-KUTTA del tablero" }}{PARA 4 "" 0 "" {TEXT 
261 12 "       0  | " }}{PARA 4 "" 0 "" {TEXT 262 15 "    1/2  |  1/2
" }}{PARA 4 "" 0 "" {TEXT 265 22 "    1/2  |     0   1/2" }}{PARA 4 "
" 0 "" {TEXT 266 29 "       1  |     0      0    1" }}{PARA 4 "" 0 "" 
{TEXT -1 12 "   ---------" }{TEXT 263 16 "----------------" }}{PARA 4 
"" 0 "" {TEXT 264 35 "          |  1/6    1/3   1/3   1/6" }}{PARA 4 "
" 0 "" {TEXT 259 73 "para aproximar la soluci\363n en  x = 0.3  tomand
o amplitud de paso  h = 0.1" }}{PARA 4 "" 0 "" {TEXT 260 2 "Re" }
{TEXT -1 63 "al\355cense las operaciones con 3 cifras significativas y
 redondeo" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{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 10 "Digits:=3:" }}}{EXCHG {PARA 4 "" 0 "" {TEXT 267 41 "El sistema
 (aut.) y la verdadera soluci\363n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 51 "f:=unapply([(-y[1]^2+y[2]^4)/y[2]^2,-y[1]/y[2]],y):" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "yv[1]:=unapply(sin(x)*cos
(x),x):\nyv[2]:=unapply(cos(x),x):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
270 61 "RK4 -> Runge-Kutta 4 evaluaciones y orden 4 cl\341sico (s. aut
.)" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 264 "R
kut4:=proc(f::procedure,yini::list,nit::posint,h::numeric)\nlocal k,k1
,k2,k3,k4:\nglobal y0:\ny0:=yini:\nfor k from 1 to nit do\nk1:=f(y0):
\nk2:=f(y0+(1./2.)*h*k1):\nk3:=f(y0+(1./2.)*h*k2);\nk4:=f(y0+h*k3);\ny
0:=y0+h*((1./6.)*k1+(1./3.)*k2+(1./3.)*k3+(1./6.)*k4):\nod:\nend:" }}}
{EXCHG {PARA 4 "" 0 "" {TEXT -1 48 "Estamos en condiciones de comenzar
 con el m\351todo" }{TEXT 272 1 " " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "x0:=0;y0:=[0.,1.];h:=0.1;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x0G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y0G7$
$\"\"!F'$\"\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG$\"\"\"!\"
\"" }}}{EXCHG {PARA 4 "" 0 "" {TEXT 271 11 "Primer paso" }{MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "x1:=x0+h;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 50 "k1:=f(y0):print(`1k1=`,k1[1]):print(`2k1=`,k1[2]
):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%1k1=G$\"$+\"!\"#" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6$%%2k1=G$!\"!\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 63 "k2:=f(y0+(1./2.)*h*k1):print(`1k2=`,k2[1]):print(`2k2
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{PARA 11 "" 1 "" {XPPMATH 20 "6$%%2k2=G$!$+&!\"%" }}}{EXCHG {PARA 0 ">
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\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%2k3=G$!$+&!\"%" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "k4:=f(y0+h*k3):print(`1k4=`,k4[1]):
print(`2k4=`,k4[2]):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%1k4=G$\"$!)*
!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%2k4=G$!$***!\"%" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "y1:=y0+h*((1./6.)*k1+(1./3.)*k2+(1.
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "k1:=f(y0):print(`1k1=`,k1[1]
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)*!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%2k1=G$!$***!\"%" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "k2:=f(y0+(1./2.)*h*k1):print
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{XPPMATH 20 "6$%%1k3=G$\"$a*!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%
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{XPPMATH 20 "6$%%1k4=G$\"$@*!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%
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{XPPMATH 20 "6#>%#y1G7$$\"$&>!\"$$\"$!)*F(" }}}{EXCHG {PARA 4 "" 0 "" 
{TEXT 269 11 "Tercer paso" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 
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{XPPMATH 20 "6$%%1k1=G$\"$@*!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%
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{PARA 11 "" 1 "" {XPPMATH 20 "6$%%1k4=G$\"$D)!\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$%%2k4=G$!$'H!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 55 "y1:=y0+h*((1./6.)*k1+(1./3.)*k2+(1./3.)*k3+(1./6.)*k4);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G7$$\"$$G!\"$$\"$b*F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "yv[1](x1)-y1[1];yv[2](x1)-y1[2];" }
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