{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 0 0 0 0 0 0 2 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 "" -1 256 "" 1 14 25 1 1 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 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 1 14 25 1 1 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 18 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 12 0 0 56 0 0 1 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 244 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 244 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE " Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 4 "" 0 "" {TEXT 259 14 "Ejercicios de " }}{PARA 4 "" 0 "" {TEXT 260 46 "(5) Sistemas aut\363nomos: m\351todos de RUNGE -KUTTA" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 258 15 "Ejercicio 05-03" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 " " {TEXT -1 33 "Para el problema de segundo orden" }}{PARA 4 "" 0 "" {TEXT -1 16 " y'' + y = 0" }}{PARA 4 "" 0 "" {TEXT -1 26 " y(0 ) = 0 , y'(0) = 1" }}{PARA 4 "" 0 "" {TEXT -1 51 "y convertido en el \+ problema tipo aut\363nomo de Cauchy" }}{PARA 4 "" 0 "" {TEXT -1 17 " \+ y'_1 = y_2 " }}{PARA 4 "" 0 "" {TEXT -1 17 " y'_2 = - y_1" }} {PARA 4 "" 0 "" {TEXT -1 28 " y_1(0) = 0 , y_2(0) = 1" }}{PARA 4 " " 0 "" {TEXT -1 104 "(de soluci\363n exacta y_1(x) = sen(x) , y_2(x) = cos(x) , o sea y(x) = sen(x) ) util\355cese el m\351todo de " }} {PARA 4 "" 0 "" {TEXT -1 23 "RUNGE-KUTTA del tablero" }}{PARA 4 "" 0 " " {TEXT 264 12 " 0 | " }}{PARA 4 "" 0 "" {TEXT 265 15 " 1/2 \+ | 1/2" }}{PARA 4 "" 0 "" {TEXT 268 22 " 1/2 | 0 1/2" }} {PARA 4 "" 0 "" {TEXT 269 29 " 1 | 0 0 1" }}{PARA 4 "" 0 "" {TEXT -1 12 " ---------" }{TEXT 266 16 "----------------" }}{PARA 4 "" 0 "" {TEXT 267 35 " | 1/6 1/3 1/3 1/6" } }{PARA 4 "" 0 "" {TEXT 256 73 "para aproximar la soluci\363n en x = 0 .3 tomando amplitud de paso h = 0.1" }}{PARA 4 "" 0 "" {TEXT 261 2 " Re" }{TEXT -1 63 "al\355cense las operaciones con 3 cifras significati vas y redondeo" }}}{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 257 41 "El sistema (aut.) y la verdadera soluci \363n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "f:=unapply([y[2],- y[1]],y):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "yv[1]:=unapply (sin(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 "Rkut4:=proc(f::procedure,yini::list,nit::posint,h::numeric)\nloca l 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 );\ny0:=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 com enzar 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 =`,k2[2]):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%1k2=G$\"\"\"\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%%2k2=G$!$+&!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "k3:=f(y0+(1./2.)*h*k2):print(`1k3=`,k3[1]):pr int(`2k3=`,k3[2]):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%1k3=G$\"$)**! \"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%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. /3.)*k3+(1./6.)*k4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G7$$\"$)* *!\"%$\"$&**!\"$" }}}{EXCHG {PARA 4 "" 0 "" {TEXT 262 12 "Segundo paso " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "y0:= y1;x1:=x1+h;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y0G7$$\"$)**!\"%$\" $&**!\"$" }}{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=`,k2[2]):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%%1k2=G$\"$!**!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%2k2=G$!$]\"! \"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "k3:=f(y0+(1./2.)*h*k 2):print(`1k3=`,k3[1]):print(`2k3=`,k3[2]):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%1k3=G$\"$))*!\"$" }}{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./3.)*k3+(1./6.)*k4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G7$$\"$*>!\"$$\"$!)*F(" }}}{EXCHG {PARA 4 "" 0 "" {TEXT 263 11 "Tercer paso" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "y0:=y1;x1:=x1+h;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y0G7$$\"$*>!\"$$\"$!)*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#x1G$\"\"$!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "k1:=f(y 0):print(`1k1=`,k1[1]):print(`2k1=`,k1[2]):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%1k1=G$\"$!)*!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% %2k1=G$!$*>!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "k2:=f(y0 +(1./2.)*h*k1):print(`1k2=`,k2[1]):print(`2k2=`,k2[2]):" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%%1k2=G$\"$q*!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%2k2=G$!$[#!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "k 3:=f(y0+(1./2.)*h*k2):print(`1k3=`,k3[1]):print(`2k3=`,k3[2]):" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%%1k3=G$\"$o*!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%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$\"$b*!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%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$$\"$'H!\"$$\"$b*F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "yv[1](x1)-y1[1];yv[2](x1)-y1[2];" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\"!F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\"!F$" }}}}{MARK "0 0 0" 9 }{VIEWOPTS 1 1 0 3 4 1802 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }