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{SECT 0 {EXCHG {PARA 4 "" 0 "" {TEXT 270 14 "Ejercicios de " }}{PARA 
4 "" 0 "" {TEXT 271 48 "(3) Ecuaciones escalares: m\351todos de RUNGE-
KUTTA" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 269 15 
"Ejercicio 03-02" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" 
{TEXT -1 68 "Consid\351rese la familia general de m\351todos de Runge-
Kutta de 2 etapas" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" 
{TEXT 256 11 "     0   | " }}{PARA 4 "" 0 "" {TEXT 257 4 "    " }
{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" }{TEXT 262 9 "   |     " }
{XPPEDIT 18 0 "a[2,1]" "6#&%\"aG6$\"\"#\"\"\"" }}{PARA 4 "" 0 "" 
{TEXT -1 14 "     ---------" }{TEXT 258 14 "--------------" }}{PARA 4 
"" 0 "" {TEXT 259 21 "           |         " }{XPPEDIT 18 0 "b[1]" "6#
&%\"bG6#\"\"\"" }{TEXT 263 6 "      " }{XPPEDIT 18 0 "b[2]" "6#&%\"bG6
#\"\"#" }{TEXT 260 3 "   " }}{PARA 4 "" 0 "" {TEXT -1 70 "B\372squense
 los m\351todos de orden 2 y de ellos el que minimiza la norma-2" }}
{PARA 4 "" 0 "" {TEXT -1 70 "de los coeficientes de las diferenciales \+
elementales del t\351rmino en   " }{XPPEDIT 18 0 "h^(3)" "6#*$%\"hG\"
\"$" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "resta
rt:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "interface(labeling=f
alse):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 265 35 "desarrollo de la verda
dera soluc\363n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "f0:=f(x
,y(x)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "f1:=subs(diff(y(
x),x)=f(x,y(x)),diff(f0,x)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 44 "f2:=subs(diff(y(x),x)=f(x,y(x)),diff(f1,x)):" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 43 "f0:=subs(\{y(x)=y0,x=x0\},f0):f0:=expand(f0
):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "f1:=subs(\{y(x)=y0,x=
x0\},f1):f1:=expand(f1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 
"f2:=subs(\{y(x)=y0,x=x0\},f2):f2:=expand(f2):" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT 266 22 "desarrollo del m\351todo " }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 12 "k10:=f(x,y):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "k20:=f(x+c[2]*h,y+a[2,1]*h*k10):" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 20 "k100:=subs(h=0,k10):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 20 "k200:=subs(h=0,k20):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 69 "Fi0:=b[1]*k100+b[2]*k200:Fi0:=subs(\{x=x0,y=y0\}
,Fi0):Fi0:=expand(Fi0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "
k11:=diff(k10,h):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "k21:=d
iff(k20,h):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "k110:=subs(h
=0,k11):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "k210:=subs(h=0,
k21):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "Fi1:=b[1]*k110+b[2
]*k210:Fi1:=subs(\{x=x0,y=y0\},Fi1):Fi1:=expand(Fi1):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "k12:=diff(k11,h):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 17 "k22:=diff(k21,h):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 20 "k120:=subs(h=0,k12):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 20 "k220:=subs(h=0,k22):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 69 "Fi2:=b[1]*k120+b[2]*k220:Fi2:=subs(\{x=x0,y=y0\},Fi
2):Fi2:=expand(Fi2):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 267 28 "condicio
nes para el orden 1 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "con
d1a:=coeff(Fi0,f(x0,y0))=coeff(f0,f(x0,y0));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'cond1aG/,&&%\"bG6#\"\"\"F*&F(6#\"\"#F*F*" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 268 42 "condiciones para el orden 2 y su solucio
n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "cond2a:=coeff(Fi1,D[1
](f)(x0,y0))=coeff((1/2)*f1,D[1](f)(x0,y0));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'cond2aG/*&&%\"bG6#\"\"#\"\"\"&%\"cGF)F+#F+F*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "cond2b:=coeff(coeff(Fi1,D[2]
(f)(x0,y0)),f(x0,y0))=coeff(coeff((1/2)*f1,D[2](f)(x0,y0)),f(x0,y0));
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond2bG/*&&%\"bG6#\"\"#\"\"\"&%
\"aG6$F*F+F+#F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "sol_or
den2:=solve(\{cond1a,cond2a,cond2b\},\{a[2,1],b[1],b[2]\});" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%+sol_orden2G<%/&%\"bG6#\"\"\",$*&#F*\"\"#F
**&,&F*!\"\"*&F.F*&%\"cG6#F.F*F*F*F3F1F*F*/&F(F5,$*&F-F**&F*F*F3F1F*F*
/&%\"aG6$F.F*F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "a[2,1]:=
subs(sol_orden2,a[2,1]);b[1]:=subs(sol_orden2,b[1]);b[2]:=subs(sol_ord
en2,b[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6$\"\"#\"\"\"&%\"
cG6#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"bG6#\"\"\",$*&#F'\"\"#F
'*&,&F'!\"\"*&F+F'&%\"cG6#F+F'F'F'F0F.F'F'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%\"bG6#\"\"#,$*&#\"\"\"F'F+*&F+F+&%\"cGF&!\"\"F+F+" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 34 "coeficientes del t\351rmino de \+
error " }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 34 "# coeficiente del \+
t\351rmino de error" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "term
:=expand((1/2!)*Fi2-(1/3!)*f2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 37 "erro3a:=coeff(term,D[1,1](f)(x0,y0));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'erro3aG,&*&#\"\"\"\"\"%F(&%\"cG6#\"\"#F(F(#F(\"\"'!
\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "erro3b:=coeff(coeff
(term,D[1,2](f)(x0,y0)),f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%'erro3bG,&*&#\"\"\"\"\"#F(&%\"cG6#F)F(F(#F(\"\"$!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "erro3c:=coeff(coeff(term,D[2,2](f)(
x0,y0)),f(x0,y0)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'erro3cG,&*&
#\"\"\"\"\"%F(&%\"cG6#\"\"#F(F(#F(\"\"'!\"\"" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 57 "erro3d:=coeff(coeff(term,D[2](f)(x0,y0)),D[1](f)
(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'erro3dG#!\"\"\"\"'" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "erro3e:=coeff(coeff(term,D[
2](f)(x0,y0)^2),f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'erro3
eG#!\"\"\"\"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 261 52 "minimizar la no
rma-2 de estos coeficientes del error" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 75 "norma2cuadrado_error:=expand(erro3a^2+erro3b^2+erro3c
^2+erro3d^2+erro3e^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%5norma2cua
drado_errorG,(*&#\"\"$\"\")\"\"\"*$)&%\"cG6#\"\"#F0F*F*F**&#F*F0F*F-F*
!\"\"#F0\"\"*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "der_norm
a2cuadrado_error:=diff(norma2cuadrado_error,c[2]);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%9der_norma2cuadrado_errorG,&*&#\"\"$\"\"%\"\"\"&%\"
cG6#\"\"#F*F*#F*F.!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "
der2_norma2cuadrado_error:=diff(der_norma2cuadrado_error,c[2]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%:der2_norma2cuadrado_errorG#\"\"$\"
\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "c2_minimo:=subs(solv
e(\{der_norma2cuadrado_error\},\{c[2]\}),c[2]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%*c2_minimoG#\"\"#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 81 "en_minimo:=subs(c[2]=c2_minimo,norma2cuadrado_error);
en_minimo:=evalf(en_minimo);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*en_
minimoG#\"\"\"\"#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*en_minimoG$\"
+cbbbb!#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "subs(c[2]=c2_m
inimo,[`c[2]`=c[2],`a[2,1]`=a[2,1],`b[1]`=b[1],`b[2]`=b[2]]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7&/%%c[2]G#\"\"#\"\"$/%'a[2,1]GF&/%%b[
1]G#\"\"\"\"\"%/%%b[2]G#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 }