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{SECT 0 {EXCHG {PARA 4 "" 0 "" {TEXT 268 14 "Ejercicios de " }}{PARA 
4 "" 0 "" {TEXT 269 48 "(3) Ecuaciones escalares: m\351todos de RUNGE-
KUTTA" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 267 15 
"Ejercicio 03-17" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" 
{TEXT -1 31 "Para el problema tipo de Cauchy" }}{PARA 4 "" 0 "" {TEXT 
-1 16 "     y' = x y + " }{XPPEDIT 18 0 "x^2" "6#*$%\"xG\"\"#" }}
{PARA 4 "" 0 "" {TEXT -1 17 "     y(0) = 1 ,  " }}{PARA 4 "" 0 "" 
{TEXT -1 21 "(de soluci\363n exacta  " }{XPPEDIT 18 0 "-x+1/2*exp(1/2*
x^2)*Pi^(1/2)*2^(1/2)*erf(1/2*2^(1/2)*x)+1.*exp(1/2*x^2)" "6#,(%\"xG!
\"\"*.\"\"\"F'\"\"#F%-%$expG6#*(F'F'F(F%F$F(F')%#PiG*&F'F'F(F%F')F(*&F
'F'F(F%F'-%$erfG6#**F'F'F(F%)F(*&F'F'F(F%F'F$F'F'F'*&-%&FloatG6$F'\"\"
!F'-F*6#*(F'F'F(F%F$F(F'F'" }{TEXT -1 39 "  ,  donde  erf  representa \+
la funci\363n " }}{PARA 4 "" 0 "" {TEXT -1 21 "de error    erf(x) = " 
}{XPPEDIT 18 0 "2/sqrt(Pi)" "6#*&\"\"#\"\"\"-%%sqrtG6#%#PiG!\"\"" }
{TEXT -1 1 " " }{XPPEDIT 18 0 " int((exp(-t^2), t=0..x);" "6#-%$intG6$
-%$expG6#,$*$%\"tG\"\"#!\"\"/F+;\"\"!%\"xG" }{TEXT -1 56 "    ) util
\355cese el m\351todo de Runge-Kutta del tablero    " }}{PARA 4 "" 0 "
" {TEXT 257 12 "       0  | " }}{PARA 4 "" 0 "" {TEXT 258 15 "    1/2 \+
 |  1/2" }}{PARA 4 "" 0 "" {TEXT 261 22 "    1/2  |     0   1/2" }}
{PARA 4 "" 0 "" {TEXT 262 29 "       1  |     0      0    1" }}{PARA 
4 "" 0 "" {TEXT -1 12 "   ---------" }{TEXT 259 16 "----------------" 
}}{PARA 4 "" 0 "" {TEXT 260 35 "          |  1/6    1/3   1/3   1/6" }
}{PARA 4 "" 0 "" {TEXT 256 73 "para aproximar la soluci\363n en  x = 0
.3  tomando amplitud de paso  h = 0.1" }}{PARA 4 "" 0 "" {TEXT 270 2 "
Re" }{TEXT -1 63 "al\355cense las operaciones con 3 cifras significati
vas y redondeo" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "interface(labeling=false)
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Digits:=3:" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 263 42 "La ecuaci\363n (aut.) y la verdadera sol
uci\363n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "f:=(x,y)->x*y+x
^2:yv:=x->-x+1/2*exp(1/2*x^2)*Pi^(1/2)*2^(1/2)*erf(1/2*2^(1/2)*x)+1.*e
xp(1/2*x^2):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 64 "RK4 -> Runge-Kut
ta 4 evaluaciones y orden 4 cl\341sico (e. no aut.)" }{MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 341 "Rkut4:=proc(f::procedu
re,ytotalini::list,nit::posint,h::numeric)\nlocal x0,y0,k,k1,k2,k3,k4:
\nx0:=ytotalini[1]:y0:=ytotalini[2]:\nfor k from 1 to nit do\nk1:=f(x0
,y0):\nk2:=f(x0+(1./2.)*h,y0+(1./2.)*h*k1):\nk3:=f(x0+(1./2.)*h,y0+(1.
/2.)*h*k2):\nk4:=f(x0+h,y0+h*k3):\nx0:=x0+h:y0:=y0+h*((1./6.)*k1+(1./3
.)*k2+(1./3.)*k3+(1./6.)*k4):\n[x0,y0]:\nod:\nend:" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT 265 11 "Primer paso" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 43 "h:=0.1:xini:=0.:yini:=1.:x0:=xini;y0:=yin
i;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G$\"\"!F&" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%#y0G$\"\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "x1:=xini+h;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G
$\"\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "k1:=f(x0,y0
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k1G$\"\"!F&" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 36 "k2:=f(x0+(1./2.)*h,y0+(1./2.)*h*k1);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k2G$\"$D&!\"%" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 36 "k3:=f(x0+(1./2.)*h,y0+(1./2.)*h*k2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k3G$\"$D&!\"%" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 20 "k4:=f(x0+h,y0+h*k3);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#k4G$\"$6\"!\"$" }}}{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#>%#y1G$\"$,\"!\"#" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 266 53 "Todos los pasos de h = 0.1 desde x = 0. \+
hasta x = 0.3" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 48 "h:=0.1;xini:=0.;yini:=1.;ytotalini:=[xini,yini];" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG$\"\"\"!\"\"" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%%xiniG$\"\"!F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%%yiniG$\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*ytotaliniG7$
$\"\"!F'$\"\"\"F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "for i
 from 1 to 3 do ytotal1:=Rkut4(f,ytotalini,1,h):print(`[x,y] = `,ytota
l1,`   mientras la exacta vale`,evalf(yv(ytotal1[1]))):ytotalini:=ytot
al1: od:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%)[x,y]~=~G7$$\"\"\"!\"\"$
\"$,\"!\"#%;~~~mientras~la~exacta~valeGF(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6&%)[x,y]~=~G7$$\"\"#!\"\"$\"$.\"!\"#%;~~~mientras~la~exa
cta~valeG$\"$-\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%)[x,y]~=~G7$$\"
\"$!\"\"$\"$1\"!\"#%;~~~mientras~la~exacta~valeGF(" }}}}{MARK "0 3 0" 
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