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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 05-05" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "
" {TEXT -1 34 "Para el problema tipo no aut\363nomo " }}{PARA 4 "" 0 "
" {TEXT -1 35 "     y'_1 =    y_1 + 2  y_2 +  x   " }}{PARA 4 "" 0 "" 
{TEXT -1 37 "     y'_2 =    y_1               - x " }}{PARA 4 "" 0 "" 
{TEXT -1 28 "     y_1(0) = 0 , y_2(0) = 0" }}{PARA 4 "" 0 "" {TEXT -1 
32 "(de soluci\363n exacta   y_1(x) =  " }{XPPEDIT 18 0 "x-1+exp(-x)" 
"6#,(%\"xG\"\"\"F%!\"\"-%$expG6#,$F$F&F%" }{TEXT -1 12 " , y_2(x) = " 
}{XPPEDIT 18 0 "-x+1-exp(-x)" "6#,(%\"xG!\"\"\"\"\"F&-%$expG6#,$F$F%F%
" }{TEXT -1 26 " ) 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 75 "para aproximar la soluci\363n en  x = 0.3  tomand
o amplitud de paso  h = 0.1  " }}{PARA 4 "" 0 "" {TEXT 272 76 "Luego s
e convertir\341 el sistema en aut\363nomo, pero de tama\361o 3, y se r
epetir\341n" }}{PARA 4 "" 0 "" {TEXT 273 45 "los c\341lculos, comparan
do los datos obtenidos." }}{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 
102 "simplify(dsolve(\{diff(y1(x),x)=y1(x)+2*y2(x)+x,diff(y2(x),x)=y1(
x)-x,y1(0)=0,y2(0)=0\},\{y1(x),y2(x)\}));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#<$/-%#y2G6#%\"xG,(\"\"\"F*-%$expG6#,$F(!\"\"F/F(F//-%#y
1GF',(F/F*F+F*F(F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "simp
lify(dsolve(\{diff(y1(x),x)=y1(x)+2*y2(x)+y3(x),diff(y2(x),x)=y1(x)-y3
(x),diff(y3(x),x)=1,y1(0)=0,y2(0)=0,y3(0)=0\},\{y1(x),y2(x),y3(x)\}));
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/-%#y2G6#%\"xG,(\"\"\"F*-%$expG6
#,$F(!\"\"F/F(F//-%#y1GF',(F/F*F+F*F(F*/-%#y3GF'F(" }}}{EXCHG {PARA 4 
"" 0 "" {TEXT 275 16 "Caso no aut\363nomo" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
26 "interface(labeling=false):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 10 "Digits:=3:" }}}{EXCHG {PARA 4 "" 0 "" {TEXT 276 44 "El sistema
 (no aut.) y la verdadera soluci\363n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "f:=unapply([y[1]+2*y[2]+x,y[1]-x],x,y):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "yv[1]:=unapply(x-1+exp(-x),x):\nyv[
2]:=unapply(-x+1-exp(-x),x):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 281 64 "
RK4 -> Runge-Kutta 4 evaluaciones y orden 4 cl\341sico (s. no aut.)" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 344 "Rkut4:
=proc(f::procedure,xini::numeric,yini::list,nit::posint,h::numeric)\nl
ocal k,k1,k2,k3,k4;\nglobal x0,y0;\nx0:=xini;\ny0:=yini;\nfor k from 1
 to nit do\nk1:=f(x0,y0);\nk2:=f(x0+(1./2.)*h,y0+(1./2.)*h*k1);\nk3:=f
(x0+(1./2.)*h,y0+(1./2.)*h*k2);\nk4:=f(x0+h,y0+h*k3);\nx0:=x0+h;\ny0:=
y0+h*((1./6.)*k1+(1./3.)*k2+(1./3.)*k3+(1./6.)*k4);\n[x0,y0];\nod:\nen
d:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "xini:=0.;yini:=[0.,0.
];solu:=Rkut4(f,xini,yini,10,0.03);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%%xiniG$\"\"!F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%yiniG7$$\"\"!F
'F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%soluG7$$\"#I!\"#7$$\"$4%!\"%
$!$4%F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "[solu[2,1],solu
[2,2]];[yv[1](solu[1]),yv[2](solu[1])];[solu[2,1]-yv[1](solu[1]),solu[
2,2]-yv[2](solu[1])];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"$4%!\"%$
!$4%F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"#T!\"$$!#TF&" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#7$$!\"\"!\"%$\"\"\"F&" }}}{EXCHG {PARA 4 "" 
0 "" {TEXT -1 48 "Estamos en condiciones de comenzar con el m\351todo
" }{TEXT 280 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "x0:=0;
y0:=[0.,0.];h:=0.1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y0G7$$\"\"!F'F&" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"hG$\"\"\"!\"\"" }}}{EXCHG {PARA 4 "" 0 "" {TEXT 
277 11 "Primer paso" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "x1:=x0+h;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"
\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "k1:=f(x0,y0):p
rint(`1k1=`,k1[1]):print(`2k1=`,k1[2]):" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6$%%1k1=G$\"\"!F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%2k1=G$\"\"!
F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "k2:=f(x0+(1./2.)*h,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 76 "k
3:=f(x0+(1./2.)*h,y0+(1./2.)*h*k2):print(`1k3=`,k3[1]):print(`2k3=`,k3
[2]):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%1k3=G$\"$v%!\"%" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6$%%2k3=G$!$v%!\"%" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 60 "k4:=f(x0+h,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$$\"$$[!\"&$!$$[F
(" }}}{EXCHG {PARA 4 "" 0 "" {TEXT 278 12 "Segundo paso" }{MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "x0:=x1:y0:=y1;x1:=
x1+h;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y0G7$$\"$$[!\"&$!$$[F(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"\"#!\"\"" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 53 "k1:=f(x0,y0):print(`1k1=`,k1[1]):print(`2k1=
`,k1[2]):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%1k1=G$\"$_*!\"%" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$%%2k1=G$!$_*!\"%" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 76 "k2:=f(x0+(1./2.)*h,y0+(1./2.)*h*k1):print(`1k
2=`,k2[1]):print(`2k2=`,k2[2]):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%1
k2=G$\"$S\"!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%2k2=G$!$S\"!\"$" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "k3:=f(x0+(1./2.)*h,y0+(1.
/2.)*h*k2):print(`1k3=`,k3[1]):print(`2k3=`,k3[2]):" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6$%%1k3=G$\"$Q\"!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
$%%2k3=G$!$Q\"!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "k4:=f
(x0+h,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$$\"$'=!\"%$!$'=F(" }}}{EXCHG {PARA 4 "" 
0 "" {TEXT 279 11 "Tercer paso" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 23 "x0:=x1:y0:=y1;x1:=x1+h;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#y0G7$$\"$'=!\"%$!$'=F(" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#x1G$\"\"$!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "
k1:=f(x0,y0):print(`1k1=`,k1[1]):print(`2k1=`,k1[2]):" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6$%%1k1=G$\"$\"=!\"$" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6$%%2k1=G$!$\"=!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "
k2:=f(x0+(1./2.)*h,y0+(1./2.)*h*k1):print(`1k2=`,k2[1]):print(`2k2=`,k
2[2]):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%1k2=G$\"$A#!\"$" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6$%%2k2=G$!$A#!\"$" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 76 "k3:=f(x0+(1./2.)*h,y0+(1./2.)*h*k2):print(`1k3=`,k3
[1]):print(`2k3=`,k3[2]):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%1k3=G$
\"$?#!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%2k3=G$!$?#!\"$" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "k4:=f(x0+h,y0+h*k3):print(`1
k4=`,k4[1]):print(`2k4=`,k4[2]):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%
1k4=G$\"$f#!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%2k4=G$!$f#!\"$" }
}}{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#>%#y1
G7$$\"$1%!\"%$!$1%F(" }}}{EXCHG {PARA 4 "" 0 "" {TEXT 283 25 "Al cabo \+
de los tres pasos" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "[y1[1]
,y1[2]];[yv[1](x1),yv[2](x1)];[y1[1]-yv[1](x1),y1[2]-yv[2](x1)];" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"$1%!\"%$!$1%F&" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#7$$\"#T!\"$$!#TF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
7$$!\"%F%$\"\"%F%" }}}{EXCHG {PARA 4 "" 0 "" {TEXT 274 13 "Caso aut
\363nomo" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "interface(labeling=false):" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Digits:=3:" }}}{EXCHG 
{PARA 4 "" 0 "" {TEXT 267 41 "El sistema (aut.) y la verdadera soluci
\363n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "f:=unapply([y[1]+2
*y[2]+y[3],y[1]-y[3],1],y):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
83 "yv[1]:=unapply(x-1+exp(-x),x):\nyv[2]:=unapply(-x+1-exp(-x),x):\ny
v[3]:=unapply(x,x):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 271 61 "RK4 -> Ru
nge-Kutta 4 evaluaciones y orden 4 cl\341sico (s. aut.)" }{MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 264 "Rkut4:=proc(f::pr
ocedure,yini::list,nit::posint,h::numeric)\nlocal k,k1,k2,k3,k4:\nglob
al y0:\ny0:=yini:\nfor k from 1 to nit do\nk1:=f(y0):\nk2:=f(y0+(1./2.
)*h*k1):\nk3:=f(y0+(1./2.)*h*k2);\nk4:=f(y0+h*k3);\ny0:=y0+h*((1./6.)*
k1+(1./3.)*k2+(1./3.)*k3+(1./6.)*k4):\nod:\nend:" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 45 "yini:=[0.,0.,0.];solu:=Rkut4(f,yini,10,0.03);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%yiniG7%$\"\"!F'F&F&" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%%soluG7%$\"$4%!\"%$!$4%F($\"$+$!\"$" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "[solu[1],solu[2],solu[3]];[
yv[1](solu[3]),yv[2](solu[3]),yv[3](solu[3])];[solu[1]-yv[1](solu[3]),
solu[2]-yv[2](solu[3]),solu[3]-yv[3](solu[3])];" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7%$\"$4%!\"%$!$4%F&$\"$+$!\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7%$\"#T!\"$$!#TF&$\"$+$F&" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#7%$!\"\"!\"%$\"\"\"F&$\"\"!F*" }}}{EXCHG {PARA 4 "" 0 "" {TEXT 
-1 48 "Estamos en condiciones de comenzar con el m\351todo" }{TEXT 
270 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "x0:=0;y0:=[0.,0
.,0.];h:=0.1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G\"\"!" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#y0G7%$\"\"!F'F&F&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"hG$\"\"\"!\"\"" }}}{EXCHG {PARA 4 "" 0 "" {TEXT 
282 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$\"
\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "k1:=f(y0):prin
t(`1k1=`,k1[1]):print(`2k1=`,k1[2]):" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6$%%1k1=G$\"\"!F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%2k1=G$\"\"!F%" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "k2:=f(y0+(1./2.)*h*k1):pr
int(`1k2=`,k2[1]):print(`2k2=`,k2[2]):" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6$%%1k2=G$\"$+&!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%2k2=G$!$+
&!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "k3:=f(y0+(1./2.)*h
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{XPPMATH 20 "6$%%1k3=G$\"$v%!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%
2k3=G$!$v%!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "k4:=f(y0+
h*k3):print(`1k4=`,k4[1]):print(`2k4=`,k4[2]):" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$%%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($\"$+\"!\"$" }}}{EXCHG {PARA 
4 "" 0 "" {TEXT 268 12 "Segundo paso" }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "y0:=y1;x1:=x1+h;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#y0G7%$\"$$[!\"&$!$$[F($\"$+\"!\"$" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#x1G$\"\"#!\"\"" }}}{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$%%1k2=G$\"$S\"!\"$" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$%%2k2=G$!$S\"!\"$" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 63 "k3:=f(y0+(1./2.)*h*k2):print(`1k3=`,k3[1]):p
rint(`2k3=`,k3[2]):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%1k3=G$\"$Q\"!
\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%2k3=G$!$Q\"!\"$" }}}{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($\"$+#!\"$" }}}{EXCHG {PARA 4 "" 0 "" {TEXT 269 11 "Tercer \+
paso" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "
y0:=y1;x1:=x1+h;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y0G7%$\"$'=!\"%
$!$'=F($\"$+#!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"\"$!\"\"
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "k1:=f(y0):print(`1k1=`,
k1[1]):print(`2k1=`,k1[2]):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%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
$%%1k2=G$\"$A#!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%2k2=G$!$A#!\"$
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "k3:=f(y0+(1./2.)*h*k2):
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):pri
nt(`1k4=`,k4[1]):print(`2k4=`,k4[2]):" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6$%%1k4=G$\"$f#!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%2k4=G$!$f#!
\"$" }}}{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%$\"$1%!\"%$!$1%F($\"$+$!\"$" }}}{EXCHG {PARA 4 "" 0 "" {TEXT 
284 25 "Al cabo de los tres pasos" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 102 "[y1[1],y1[2],y1[3]];[yv[1](x1),yv[2](x1),yv[3](x1)];
[y1[1]-yv[1](x1),y1[2]-yv[2](x1),y1[3]-yv[3](x1)];" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#7%$\"$1%!\"%$!$1%F&$\"$+$!\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7%$\"#T!\"$$!#TF&$\"\"$!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7%$!\"%F%$\"\"%F%$\"\"!F)" }}}}{MARK "0 0 0" 4 }
{VIEWOPTS 1 1 0 3 4 1802 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }