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-1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 4 "" 0 "" {TEXT 261 14 "Ejercicios de " }}{PARA 4 "" 0 "" {TEXT 262 48 "(3) Ecuaciones escalares: m\351todos de RUNGE- KUTTA" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 260 15 "Ejercicio 03-16" }}{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$ -%$expG6#,$*$%\"tG\"\"#!\"\"/F+;\"\"!%\"xG" }{TEXT -1 46 " ) util \355cese el m\351todo de TAYLOR de orden 3" }}{PARA 4 "" 0 "" {TEXT 256 73 "para aproximar la soluci\363n en x = 0.3 tomando amplitud de paso h = 0.1" }}{PARA 4 "" 0 "" {TEXT 263 2 "Re" }{TEXT -1 63 "al \355cense las operaciones con 3 cifras significativas y redondeo" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 26 "interface(labeling=false):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Digits:=3:" }}}{EXCHG {PARA 4 "" 0 "" {TEXT 257 42 "La ecuaci\363n (aut.) y la verdadera soluci\363n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "f:=(x,y)->x*y+x^2:yv:=x->-x+ 1/2*exp(1/2*x^2)*Pi^(1/2)*2^(1/2)*erf(1/2*2^(1/2)*x)+1.*exp(1/2*x^2): " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 258 37 "Los m\351todos de TAYLOR y e l de orden 3" }}}{EXCHG {PARA 4 "" 0 "" {TEXT 264 59 "Los m\351todos d e TAYLOR para los primeros \363rdenes se escriben" }}{PARA 4 "" 0 "" {TEXT 265 4 " " }{XPPEDIT 18 0 "y[n+1]" "6#&%\"yG6#,&%\"nG\"\"\"F(F (" }{TEXT 266 4 " = " }{XPPEDIT 18 0 "y[n]" "6#&%\"yG6#%\"nG" }{TEXT 267 8 " + h * " }{XPPEDIT 18 0 "T^[1]" "6#)%\"TG7#\"\"\"" }{TEXT 268 4 " ( " }{XPPEDIT 18 0 "x[n]" "6#&%\"xG6#%\"nG" }{TEXT 269 3 " , " } {XPPEDIT 18 0 "y[n]" "6#&%\"yG6#%\"nG" }{TEXT 270 34 " , h ) \+ " }}{PARA 4 "" 0 "" {TEXT 271 3 " " }{XPPEDIT 18 0 "y[n+1]" "6#&%\"yG6#,&%\"nG\"\"\"F(F(" }{TEXT 272 4 " = " } {XPPEDIT 18 0 "y[n]" "6#&%\"yG6#%\"nG" }{TEXT 273 8 " + h * " } {XPPEDIT 18 0 "T^[2]" "6#)%\"TG7#\"\"#" }{TEXT 274 4 " ( " }{XPPEDIT 18 0 "x[n]" "6#&%\"xG6#%\"nG" }{TEXT 275 3 " , " }{XPPEDIT 18 0 "y[n] " "6#&%\"yG6#%\"nG" }{TEXT 276 32 " , h ) " } }{PARA 4 "" 0 "" {TEXT 277 3 " " }{XPPEDIT 18 0 "y[n+1]" "6#&%\"yG6# ,&%\"nG\"\"\"F(F(" }{TEXT 278 4 " = " }{XPPEDIT 18 0 "y[n]" "6#&%\"yG 6#%\"nG" }{TEXT 279 8 " + h * " }{XPPEDIT 18 0 "T^[3]" "6#)%\"TG7#\" \"$" }{TEXT 282 4 " ( " }{XPPEDIT 18 0 "x[n]" "6#&%\"xG6#%\"nG" } {TEXT 280 3 " , " }{XPPEDIT 18 0 "y[n]" "6#&%\"yG6#%\"nG" }{TEXT 281 15 " , h ) " }}{PARA 4 "" 0 "" {TEXT 283 8 "donde " } {XPPEDIT 18 0 "T^[1]" "6#)%\"TG7#\"\"\"" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "T^[2]" "6#)%\"TG7#\"\"#" }{TEXT -1 3 " y " }{XPPEDIT 18 0 "T^[3] " "6#)%\"TG7#\"\"$" }{TEXT -1 3 " " }{TEXT 284 27 "representan respe ctivamente" }}{PARA 4 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "T^[1]" "6#)%\"TG7#\"\"\"" }{TEXT 288 4 " ( " }{XPPEDIT 18 0 "x[n]" "6#&%\"xG 6#%\"nG" }{TEXT 289 3 " , " }{XPPEDIT 18 0 "y[n]" "6#&%\"yG6#%\"nG" } {TEXT 290 12 " , h ) = " }{XPPEDIT 18 0 "f^[0]" "6#)%\"fG7#\"\"!" } {TEXT 285 4 " ( " }{XPPEDIT 18 0 "x[n]" "6#&%\"xG6#%\"nG" }{TEXT 286 3 " , " }{XPPEDIT 18 0 "y[n]" "6#&%\"yG6#%\"nG" }{TEXT 287 4 " ) " }} {PARA 4 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "T^[2]" "6#)%\"TG7#\" \"#" }{TEXT 294 4 " ( " }{XPPEDIT 18 0 "x[n]" "6#&%\"xG6#%\"nG" } {TEXT 295 3 " , " }{XPPEDIT 18 0 "y[n]" "6#&%\"yG6#%\"nG" }{TEXT 296 12 " , h ) = " }{XPPEDIT 18 0 "f^[0]" "6#)%\"fG7#\"\"!" }{TEXT 291 4 " ( " }{XPPEDIT 18 0 "x[n]" "6#&%\"xG6#%\"nG" }{TEXT 292 3 " , " } {XPPEDIT 18 0 "y[n]" "6#&%\"yG6#%\"nG" }{TEXT 293 6 " ) + " } {XPPEDIT 18 0 "h/2" "6#*&%\"hG\"\"\"\"\"#!\"\"" }{TEXT -1 1 " " } {TEXT 300 1 " " }{XPPEDIT 18 0 "f^[1]" "6#)%\"fG7#\"\"\"" }{TEXT 297 4 " ( " }{XPPEDIT 18 0 "x[n]" "6#&%\"xG6#%\"nG" }{TEXT 298 3 " , " } {XPPEDIT 18 0 "y[n]" "6#&%\"yG6#%\"nG" }{TEXT 299 3 " ) " }}{PARA 4 " " 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "T^[3]" "6#)%\"TG7#\"\"$" } {TEXT 304 4 " ( " }{XPPEDIT 18 0 "x[n]" "6#&%\"xG6#%\"nG" }{TEXT 305 3 " , " }{XPPEDIT 18 0 "y[n]" "6#&%\"yG6#%\"nG" }{TEXT 306 12 " , h ) = " }{XPPEDIT 18 0 "f^[0]" "6#)%\"fG7#\"\"!" }{TEXT 301 4 " ( " } {XPPEDIT 18 0 "x[n]" "6#&%\"xG6#%\"nG" }{TEXT 302 3 " , " }{XPPEDIT 18 0 "y[n]" "6#&%\"yG6#%\"nG" }{TEXT 303 6 " ) + " }{XPPEDIT 18 0 "h/ 2" "6#*&%\"hG\"\"\"\"\"#!\"\"" }{TEXT -1 1 " " }{TEXT 310 1 " " } {XPPEDIT 18 0 "f^[1]" "6#)%\"fG7#\"\"\"" }{TEXT 307 4 " ( " } {XPPEDIT 18 0 "x[n]" "6#&%\"xG6#%\"nG" }{TEXT 308 3 " , " }{XPPEDIT 18 0 "y[n]" "6#&%\"yG6#%\"nG" }{TEXT 309 6 " ) + " }{XPPEDIT 18 0 "h^ 2/6" "6#*&%\"hG\"\"#\"\"'!\"\"" }{TEXT -1 1 " " }{TEXT 314 1 " " } {XPPEDIT 18 0 "f^[2]" "6#)%\"fG7#\"\"#" }{TEXT 311 4 " ( " }{XPPEDIT 18 0 "x[n]" "6#&%\"xG6#%\"nG" }{TEXT 312 3 " , " }{XPPEDIT 18 0 "y[n] " "6#&%\"yG6#%\"nG" }{TEXT 313 3 " ) " }}{PARA 4 "" 0 "" {TEXT 315 7 " y las " }{XPPEDIT 18 0 "f^[k]" "6#)%\"fG7#%\"kG" }{TEXT 316 63 " s on las pseudoderivadas de f(x) dadas (las primeras) por" }}{PARA 4 "" 0 "" {TEXT -1 4 " " }{TEXT 317 2 " " }{XPPEDIT 18 0 "f^[0]" "6# )%\"fG7#\"\"!" }{TEXT 318 4 " = " }{XPPEDIT 18 0 "f" "6#%\"fG" }} {PARA 4 "" 0 "" {TEXT -1 4 " " }{TEXT 319 2 " " }{XPPEDIT 18 0 "f^ [1]" "6#)%\"fG7#\"\"\"" }{TEXT 320 4 " = " }{XPPEDIT 18 0 "f[x]" "6#& %\"fG6#%\"xG" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT -1 2 " " }{XPPEDIT 18 0 "f[y]" "6#&%\"fG6#%\"yG" }{TEXT -1 2 " " }} {PARA 4 "" 0 "" {TEXT -1 4 " " }{TEXT 321 2 " " }{XPPEDIT 18 0 "f^ [2]" "6#)%\"fG7#\"\"#" }{TEXT 322 4 " = " }{XPPEDIT 18 0 "f[xx]" "6#& %\"fG6#%#xxG" }{TEXT -1 5 " + 2 " }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT -1 2 " " }{XPPEDIT 18 0 "f[xy]" "6#&%\"fG6#%#xyG" }{TEXT -1 3 " + " } {XPPEDIT 18 0 "f[x]" "6#&%\"fG6#%\"xG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "f[y]" "6#&%\"fG6#%\"yG" }{TEXT -1 4 " + " }{XPPEDIT 18 0 "(f)^2" "6# *$%\"fG\"\"#" }{TEXT -1 1 " " }{XPPEDIT 18 0 "f[yy]" "6#&%\"fG6#%#yyG " }{TEXT -1 3 " + " }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT -1 1 " " } {XPPEDIT 18 0 "(f[y])^2" "6#*$&%\"fG6#%\"yG\"\"#" }{TEXT -1 1 " " }} {PARA 4 "" 0 "" {TEXT 323 63 "El \372ltimo de los m\351todos de TAYLOR escritos es el de orden 3. " }{XPPEDIT 18 0 "T^[3]" "6#)%\"TG7#\"\"$ " }{TEXT 324 34 " es justamente el desarrollo de " }{XPPEDIT 18 0 "D elta" "6#%&DeltaG" }{TEXT 325 4 " ( " }{XPPEDIT 18 0 "x[n]" "6#&%\"xG 6#%\"nG" }{TEXT 326 3 " , " }{XPPEDIT 18 0 "y[n]" "6#&%\"yG6#%\"nG" } {TEXT 327 13 " , h ) hasta" }}{PARA 4 "" 0 "" {TEXT 328 18 "dejar el \+ resto en " }{XPPEDIT 18 0 "O(h^3)" "6#-%\"OG6#*$%\"hG\"\"$" }{TEXT 329 2 " " }}}{EXCHG {PARA 4 "" 0 "" {TEXT 330 41 "Diferenciales eleme ntales que intervienen" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 68 "fx:=unapply(diff(f(x,y),x),(x,y));fy:=unapply(diff( f(x,y),y),(x,y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fxGf*6$%\"xG% \"yG6\"6$%)operatorG%&arrowGF),&9%\"\"\"*&\"\"#F/9$F/F/F)F)F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fyGf*6$%\"xG%\"yG6\"6$%)operatorG%& arrowGF)9$F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "fxx:=u napply(diff(fx(x,y),x),(x,y));fxy:=unapply(diff(fx(x,y),y),(x,y));fyy: =unapply(diff(fy(x,y),y),(x,y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% $fxxG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$fxyG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$fyyG\"\"!" }}}{EXCHG {PARA 4 "" 0 "" {TEXT 259 11 "Primer paso" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "h:=0.1:x0:=0.;y0:=1.;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G$\"\"!F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y0 G$\"\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 170 "printf(\" f= %a , fx= %a fy= %a , fxx= \+ %a fxy= %a , fyy= %a \\n\",f(x0,y0),fx(x0,y0),fy(x0,y0),fxx(x0,y0) ,fxy(x0,y0),fyy(x0,y0));" }}{PARA 6 "" 1 "" {TEXT -1 92 " \+ f= 0. , fx= 1. fy= 0. , fxx= 2 fxy= 1 \+ , fyy= 0 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "f0:=f(x0,y0); f1:=fx(x0,y0)+f(x0,y0)*fy(x0,y0);f2:=fxx(x0,y0)+2*f(x0,y0)*fxy(x0,y0)+ fx(x0,y0)*fy(x0,y0)+f(x0,y0)^2*fyy(x0,y0)+f(x0,y0)*fy(x0,y0)^2;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f0G$\"\"!F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1G$\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#f2G$\"\"#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "T3:=f0+ (h/2)*f1+(h^2/6)*f2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#T3G$\"$L&! \"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "y1:=y0+h*T3;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G$\"$,\"!\"#" }}}{EXCHG {PARA 4 " " 0 "" {TEXT 331 12 "Segundo paso" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "x0:=x0+h;y0:=y1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G$\"\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#y0G$\"$,\"!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 170 "printf (\" f= %a , fx= %a fy= %a , fx x= %a fxy= %a , fyy= %a \\n\",f(x0,y0),fx(x0,y0),fy(x0,y0),fxx(x0, y0),fxy(x0,y0),fyy(x0,y0));" }}{PARA 6 "" 1 "" {TEXT -1 96 " \+ f= .111 , fx= 1.21 fy= .1 , fxx= 2 fx y= 1 , fyy= 0 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "f0:=f( x0,y0);f1:=fx(x0,y0)+f(x0,y0)*fy(x0,y0);f2:=fxx(x0,y0)+2*f(x0,y0)*fxy( x0,y0)+fx(x0,y0)*fy(x0,y0)+f(x0,y0)^2*fyy(x0,y0)+f(x0,y0)*fy(x0,y0)^2; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f0G$\"$6\"!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1G$\"$A\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f2G$\"$M#!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "T3:= f0+(h/2)*f1+(h^2/6)*f2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#T3G$\"$w \"!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "y1:=y0+h*T3;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G$\"$.\"!\"#" }}}{EXCHG {PARA 4 " " 0 "" {TEXT 332 11 "Tercer paso" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "x0:=x0+h;y0:=y1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G$\"\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #y0G$\"$.\"!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 170 "printf( \" f= %a , fx= %a fy= %a , fxx = %a fxy= %a , fyy= %a \\n\",f(x0,y0),fx(x0,y0),fy(x0,y0),fxx(x0,y 0),fxy(x0,y0),fyy(x0,y0));" }}{PARA 6 "" 1 "" {TEXT -1 96 " \+ f= .246 , fx= 1.43 fy= .2 , fxx= 2 fxy = 1 , fyy= 0 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "f0:=f(x 0,y0);f1:=fx(x0,y0)+f(x0,y0)*fy(x0,y0);f2:=fxx(x0,y0)+2*f(x0,y0)*fxy(x 0,y0)+fx(x0,y0)*fy(x0,y0)+f(x0,y0)^2*fyy(x0,y0)+f(x0,y0)*fy(x0,y0)^2; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f0G$\"$Y#!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1G$\"$[\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#f2G$\"$z#!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "T3:=f0 +(h/2)*f1+(h^2/6)*f2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#T3G$\"$D$! \"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "y1:=y0+h*T3;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G$\"$1\"!\"#" }}}}{MARK "0 3 0" 15 }{VIEWOPTS 1 1 0 3 4 1802 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }