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2 2 0 1 } {PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 4 4 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -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 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 4 "" 0 "" {TEXT 265 14 "Ejercicios de " }}{PARA 4 "" 0 "" {TEXT 266 41 "(4) Estimaci\363n del error y cambio de paso" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 264 15 "Ejerci cio 04-01" }}{PARA 4 "" 0 "" {TEXT -1 65 "\nSe considera el par encaja do RK1(2) dado por el m\351todo de EULER" }}{PARA 4 "" 0 "" {TEXT 285 12 " 0 | " }}{PARA 4 "" 0 "" {TEXT 286 1 " " }{TEXT -1 12 " ---------" }{TEXT 287 2 "--" }}{PARA 4 "" 0 "" {TEXT 288 18 " \+ | 1" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 16 "y por el m\351todo " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 258 13 " 0 | " }}{PARA 4 "" 0 "" {TEXT 259 6 " \+ " }{XPPEDIT 18 0 "2/3" "6#*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT 262 8 " \+ | " }{XPPEDIT 18 0 "2/3" "6#*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT 263 2 " " }}{PARA 4 "" 0 "" {TEXT -1 13 " ---------" }{TEXT 260 11 "-- ---------" }}{PARA 4 "" 0 "" {TEXT 261 18 " | " } {XPPEDIT 18 0 "1/4" "6#*&\"\"\"F$\"\"%!\"\"" }{TEXT 289 6 " " } {XPPEDIT 18 0 "3/4" "6#*&\"\"$\"\"\"\"\"%!\"\"" }}{PARA 4 "" 0 "" {TEXT -1 75 "a) Se comprobar\341 que es un par encajado correctamente \+ forjado, diciendo los" }}{PARA 4 "" 0 "" {TEXT -1 68 "\363rdenes que s e emplean en efecto y las evaluaciones totales que usa." }}{PARA 4 "" 0 "" {TEXT -1 68 "Se escribir\341 el conjunto del par o en la forma de l cuadro habitual. " }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 " " {TEXT -1 61 "b) Seguidamente, se emplear\341 el par para integrar el problema" }}{PARA 4 "" 0 "" {TEXT -1 9 " y' = " }{XPPEDIT 18 0 "y^ 2" "6#*$%\"yG\"\"#" }{TEXT -1 14 " , y(0) = 1 ," }}{PARA 4 "" 0 "" {TEXT 267 18 "de soluci\363n exacta" }}{PARA 4 "" 0 "" {TEXT 290 11 " \+ y(x) = " }{XPPEDIT 18 0 "1/(1-x)" "6#*&\"\"\"F$,&F$F$%\"xG!\"\"F'" }{TEXT 309 2 " " }}{PARA 4 "" 0 "" {TEXT 291 58 "para una tolerancia \+ de E = 0.5 en el intervalo [0,1] . " }}{PARA 4 "" 0 "" {TEXT 292 44 "Se dar\341n \372nicamente los tres primeros pasos." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 315 55 "c) Tambi\351n se em plear\341 el par para integrar el problema" }}{PARA 4 "" 0 "" {TEXT -1 12 " y' = - " }{XPPEDIT 18 0 "y^2" "6#*$%\"yG\"\"#" }{TEXT -1 14 " , y(0) = 1 ," }}{PARA 4 "" 0 "" {TEXT 310 18 "de soluci\363n exa cta" }}{PARA 4 "" 0 "" {TEXT 311 11 " y(x) = " }{XPPEDIT 18 0 "1/(x +1)" "6#*&\"\"\"F$,&%\"xGF$F$F$!\"\"" }{TEXT 314 2 " " }}{PARA 4 "" 0 "" {TEXT 312 58 "para una tolerancia de E = 0.5 en el intervalo [ 0,1] . " }}{PARA 4 "" 0 "" {TEXT 313 33 "Aqu\355 ser\341n pocos los pa sos a dar." }}}{EXCHG {PARA 4 "" 0 "" {TEXT 316 2 "a)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 26 "interface(labeling=false):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 282 41 "Comprobaci\363n del orden del segundo m\351todo" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "y1:=f(x,y(x)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "y2:=subs(diff(y(x),x)=y1,diff(y1,x) ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Y1:=subs(\{y(x)=y0,x= x0\},y1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Y2:=subs(\{y(x )=y0,x=x0\},y2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "k1_:=f( x0,y0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "k2_:=f(x0+(2/3)* h,y0+(2/3)*h*k1_):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "k1_0: =subs(h=0,k1_):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "k2_0:=su bs(h=0,k2_):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "k11:=diff(k 1_,h):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "k21:=diff(k2_,h): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "k110:=subs(h=0,k11):" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "k210:=subs(h=0,k21):" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 256 27 "condiciones para el orden 1" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Fi_0:=expand((1/4)*k1_0+(3/4 )*k2_0+bt[3]*k3_0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Fi_0G,&-%\"f G6$%#x0G%#y0G\"\"\"*&&%#btG6#\"\"$F+%%k3_0GF+F+" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 48 "cond1a:=coeff(Y1,f(x0,y0))=coeff(Fi_0,f(x0,y0) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond1aG/\"\"\"F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 27 "condiciones para el orden 2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Fi10:=expand((1/4)*k110+(3/4)*k210+ bt[3]*k310);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Fi10G,(*&#\"\"\"\" \"#F(--&%\"DG6#F(6#%\"fG6$%#x0G%#y0GF(F(*&F'F(*&--&F-6#F)F/F1F(-F0F1F( F(F(*&&%#btG6#\"\"$F(%%k310GF(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "cond2a:=(1/2)*coeff(Y2,D[1](f)(x0,y0))=coeff(Fi10,D[1 ](f)(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond2aG/#\"\"\"\" \"#F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "cond2b:=(1/2)*coef f(coeff(Y2,D[2](f)(x0,y0)),f(x0,y0))=coeff(coeff(Fi10,D[2](f)(x0,y0)), f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond2bG/#\"\"\"\"\"#F &" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 71 "Por lo tanto hemos visto que el segundo m\351todo es de orden 2 (de hecho " }}{PARA 4 "" 0 "" {TEXT -1 61 "es el m\351todo de HEUN de 2 etapas) y hemos obtenido el \+ tablero" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 268 24 " 0 | " }}{PARA 4 "" 0 "" {TEXT 275 16 " \+ " }{XPPEDIT 18 0 "2/3" "6#*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT 276 10 " | " }{XPPEDIT 18 0 "2/3" "6#*&\"\"#\"\"\"\"\"$!\"\"" } {TEXT 277 2 " " }{TEXT -1 1 " " }}{PARA 4 "" 0 "" {TEXT -1 13 " -- -------" }{TEXT 269 23 "-----------------------" }}{PARA 4 "" 0 "" {TEXT 270 39 " ord 1 | 1 0" }}{PARA 0 "" 0 "" {TEXT -1 14 " ---------" }{TEXT 271 25 "----------------------- --" }}{PARA 4 "" 0 "" {TEXT 272 28 " ord 2 | " } {XPPEDIT 18 0 "1/4" "6#*&\"\"\"F$\"\"%!\"\"" }{TEXT 278 8 " " } {XPPEDIT 18 0 "3/4" "6#*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT 279 3 " " }} {PARA 0 "" 0 "" {TEXT -1 14 " ---------" }{TEXT 273 25 "---------- ---------------" }}{PARA 4 "" 0 "" {TEXT 274 24 " estim \+ | " }{XPPEDIT 18 0 "-3/4" "6#,$*&\"\"$\"\"\"\"\"%!\"\"F(" }{TEXT 280 8 " " }{XPPEDIT 18 0 "3/4" "6#*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT 281 4 " " }}}{EXCHG {PARA 4 "" 0 "" {TEXT 317 19 "b) el problema y' =" }{TEXT 319 1 " " }{XPPEDIT 18 0 "y^2" "6#*$%\"yG\"\"#" }{TEXT 318 13 " , y(0) = 1 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "interface(labeling=fal se):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Digits:=3:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 283 66 "La ecuaci\363n (aut.) y su soluci \363n; valores iniciales; la tolerancia" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "f:=y->y^2;yv:=x->1/(1-x);xini:=0;yini:=1;E:=0.5;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"yG6\"6$%)operatorG%&arrow GF(*$)9$\"\"#\"\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#yvGf*6 #%\"xG6\"6$%)operatorG%&arrowGF(*&\"\"\"F-,&F-F-9$!\"\"F0F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%xiniG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%yiniG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"E G$\"\"&!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 295 6 "El par" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "a[1,2]:=2/3;b[1]:=1;b[2]:=0; bt[1]:=1/4;bt[2]:=3/4;d[1]:=bt[1]-b[1];d[2]:=bt[2]-b[2];" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%\"aG6$\"\"\"\"\"##F(\"\"$" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>&%\"bG6#\"\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >&%\"bG6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#btG6#\"\"\" #F'\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#btG6#\"\"##\"\"$\"\"% " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"dG6#\"\"\"#!\"$\"\"%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"dG6#\"\"##\"\"$\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 294 17 "Tolerancia unidad" }{MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "TU:=E/(1-0);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#TUG$\"\"&!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 284 64 "C\341lculo de las 2 etapas, el valor de y1, y el conse jo sobre " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT 293 3 " = " } {XPPEDIT 18 0 "((TU*h)/(abs(yt[1]-y[1])))^(1.)" "6#)*(%#TUG\"\"\"%\"hG F&-%$absG6#,&&%#ytG6#F&F&&%\"yG6#F&!\"\"F2-%&FloatG6$F&\"\"!" }{TEXT 296 3 " = " }{XPPEDIT 18 0 "((TU*h)/(abs(est)))" "6#*(%#TUG\"\"\"%\"hG F%-%$absG6#%$estG!\"\"" }{TEXT 308 5 " " }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 297 32 "paso 1 con paso inicial h = 0.5 " }{TEXT 298 1 " " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 199 "x0:=xini;y0:=yini;h:=solve(h^2*abs(f(y0))=TU*h,h)[2] ;x1:=x0+h;k1:=f(y0);k2:=f(y0+a[1,2]*h*k1);y1:=y0+h*(b[1]*k1+b[2]*k2);y t1:=y0+h*(bt[1]*k1+bt[2]*k2);est:=h*(d[1]*k1+d[2]*k2);alpha:=(TU*h)/ab s(est);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y0G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"hG$\"$+&!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"$+&!\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k1G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k2G$\"$x\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #y1G$\"$]\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$yt1G$\"$z\"!\"# " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$estG$\"$!H!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG$\"$i)!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 299 61 "el paso de h = 0.5 no se acepta, en su lugar se toma \+ h = " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT 300 10 "*h = 0.431" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "h:=alpha*h;x1:=x0+h;k1:=f (y0);k2:=f(y0+a[1,2]*h*k1);y1:=y0+h*(b[1]*k1+b[2]*k2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG$\"$J%!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#x1G$\"$J%!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k1G\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k2G$\"$m\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G$\"$V\"!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "verdadero_error:=yv(x1)-y1;error_admitido:=TU*(x1-xin i);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0verdadero_errorG$\"#L!\"#" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/error_admitidoG$\"$;#!\"$" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 301 7 "paso 2 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "x0:=x0+h;y0:=y1;x1:=x0+h;k1:=f(y0);k2:=f(y0+a[1,2 ]*h*k1);y1:=y0+h*(b[1]*k1+b[2]*k2);yt1:=y0+h*(bt[1]*k1+bt[2]*k2);est:= h*(d[1]*k1+d[2]*k2);alpha:=(TU*h)/abs(est);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G$\"$J%!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# y0G$\"$V\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"$i)!\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k1G$\"$/#!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k2G$\"$3%!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# y1G$\"$J#!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$yt1G$\"$(H!\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$estG$\"$f'!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG$\"$G$!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 302 63 "el paso de h = 0.431 no se acepta, en su lugar se toma h = \+ " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT 303 10 "*h = 0.141" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "h:=alpha*h;x1:=x0+h;k1:=f(y0 );k2:=f(y0+a[1,2]*h*k1);y1:=y0+h*(b[1]*k1+b[2]*k2);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"hG$\"$T\"!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#x1G$\"$s&!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k1G$\"$/#!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k2G$\"$i#!\"#" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#y1G$\"$s\"!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "verdadero_error:=yv(x1)-y1;error_admitido:=TU*(x1-xin i);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0verdadero_errorG$\"#i!\"#" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/error_admitidoG$\"$'G!\"$" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 304 7 "paso 3 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "x0:=x0+h;y0:=y1;x1:=x0+h;k1:=f(y0);k2:=f(y0+a[1,2 ]*h*k1);y1:=y0+h*(b[1]*k1+b[2]*k2);yt1:=y0+h*(bt[1]*k1+bt[2]*k2);est:= h*(d[1]*k1+d[2]*k2);alpha:=(TU*h)/abs(est);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G$\"$s&!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# y0G$\"$s\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"$8(!\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k1G$\"$'H!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k2G$\"$+%!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# y1G$\"$9#!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$yt1G$\"$D#!\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$estG$\"$5\"!\"$" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%&alphaG$\"$T'!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 305 63 "el paso de h = 0.141 no se acepta, en su lugar se toma h = \+ " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT 307 11 "*h = 0.0904" } {TEXT 306 1 " " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "h:=alpha*h;x1:=x0+h;k1:=f(y0);k2:=f(y0+a[1,2]*h*k1);y 1:=y0+h*(b[1]*k1+b[2]*k2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG$ \"$/*!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"$i'!\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k1G$\"$'H!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k2G$\"$h$!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# y1G$\"$*>!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "verdadero_ error:=yv(x1)-y1;error_admitido:=TU*(x1-xini);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0verdadero_errorG$\"#(*!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/error_admitidoG$\"$J$!\"$" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 337 43 "comportamiento en [0,1] de la soluci\363n " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(yv(x),x=0..1);" }} {PARA 13 "" 1 "" {GLPLOT2D 496 496 496 {PLOTDATA 2 "6%-%'CURVESG6$7hp7 $$\"\"!F)$\"\"\"F)7$$\"3+++++++!=#!#>$\"3n!Q=6$eGA5!#<7$$\"3I++++++!3% F/$\"3-r^?Ya`U5F27$$\"3'**************>'F/$\"3;aM-\"3)4m5F27$$\"3?++++ ++S$)F/$\"3S$eAbV))44\"F27$$\"3'*************\\5!#=$\"3))*=adV=t6\"F27 $$\"3+++++++]7FE$\"3zUr&G9dG9\"F27$$\"3!*************\\9FE$\"3u`[FV1fp 6F27$$\"33++++++g;FE$\"3?\"4'Qn2/*>\"F27$$\"3+++++++q=FE$\"3A,I7+B,I7F 27$$\"3!*************z?FE$\"3aiiiiiii7F27$$\"35++++++!H#FE$\"3!f>>7'o, (H\"F27$$\"3++++++++DFE$\"3ELLLLLLL8F27$$\"3;+++++++FFE$\"31I')p8I')p8 F27$$\"3\")************4HFE$\"39=IaNsV59F27$$\"3+++++++?JFE$\"3SBI4s$) [`9F27$$\"3=++++++ILFE$\"3g$f7[P]#*\\\"F27$$\"3\")************RNFE$\"3 5r!*4;w)za\"F27$$\"3**************RPFE$\"35poX*3Wuf\"F27$$\"3=++++++]R 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\"3_************\\%*FE$\"3d;=======Fey7$$\"3a*************\\*FE$\"3A)* ************>Fey7$$\"3i************p&*FE$\"3IN)[`R\"eDBFey7$$\"3k***** ********f*FE$\"3)y************\\#Fey7$$\"3q************\\'*FE$\"3YaG9d G9dGFey7$$\"3u*************p*FE$\"3tILLLLLLLFeyF^\\lF^\\lF^\\lF^\\lF^ \\lF^\\lF^\\lF^\\lF^\\lF^\\l7$$\"3y************\\(*FE$\"3W'*********** **RFeyFc\\lFc\\lFc\\lFc\\lFc\\l7$$\"3#)************z(*FE$\"3yTXXXXXXXF ey7$$\"3#)*************z*FE$\"3u&*************\\FeyF]]lF]]lF]]lF]]lF]] lF]]l7$$\"3))************\\)*FE$\"3YhmmmmmmmFey7$$\"3!***************) *FE$\"29***************!#:Fg]lFg]lFg]lFg]lFg]lFg]lFg]lFg]lFg]lFg]l7$$ \"3'*************\\**FE$\"3G)*************>F\\^l7$%*undefinedGFc^l-%'C OLOURG6&%$RGBG$\"*++++\"!\")F(F(-%+AXESLABELSG6$%\"xGQ!6\"-%%VIEWG6$;F (F*%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 4 "" 0 "" {TEXT 321 19 "c) el problema y' =" }{TEXT 323 3 " - " }{XPPEDIT 18 0 "y^2" "6#*$%\"yG\"\"#" }{TEXT 322 13 " , y(0) = 1 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "res tart:" }}}{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 320 66 "La ecuaci\363n (aut.) y su soluci \363n; valores iniciales; la tolerancia" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "f:=y->-y^2;yv:=x->1/(x+1);xini:=0;yini:=1;E:=0.5;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"yG6\"6$%)operatorG%&arrow GF(,$*$)9$\"\"#\"\"\"!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #yvGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&\"\"\"F-,&9$F-F-F-!\"\"F(F(F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%xiniG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%yiniG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"E G$\"\"&!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 327 6 "El par" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "a[1,2]:=2/3;b[1]:=1;b[2]:=0; bt[1]:=1/4;bt[2]:=3/4;d[1]:=bt[1]-b[1];d[2]:=bt[2]-b[2];" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%\"aG6$\"\"\"\"\"##F(\"\"$" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>&%\"bG6#\"\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >&%\"bG6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#btG6#\"\"\" #F'\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#btG6#\"\"##\"\"$\"\"% " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"dG6#\"\"\"#!\"$\"\"%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"dG6#\"\"##\"\"$\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 326 17 "Tolerancia unidad" }{MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "TU:=E/(1-0);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#TUG$\"\"&!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 324 64 "C\341lculo de las 2 etapas, el valor de y1, y el conse jo sobre " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT 325 3 " = " } {XPPEDIT 18 0 "((TU*h)/(abs(yt[1]-y[1])))^(1.)" "6#)*(%#TUG\"\"\"%\"hG F&-%$absG6#,&&%#ytG6#F&F&&%\"yG6#F&!\"\"F2-%&FloatG6$F&\"\"!" }{TEXT 328 3 " = " }{XPPEDIT 18 0 "((TU*h)/(abs(est)))" "6#*(%#TUG\"\"\"%\"hG F%-%$absG6#%$estG!\"\"" }{TEXT 332 5 " " }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 329 32 "paso 1 con paso inicial h = 0.5 " }{TEXT 330 1 " " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 199 "x0:=xini;y0:=yini;h:=solve(h^2*abs(f(y0))=TU*h,h)[2] ;x1:=x0+h;k1:=f(y0);k2:=f(y0+a[1,2]*h*k1);y1:=y0+h*(b[1]*k1+b[2]*k2);y t1:=y0+h*(bt[1]*k1+bt[2]*k2);est:=h*(d[1]*k1+d[2]*k2);alpha:=(TU*h)/ab s(est);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y0G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"hG$\"$+&!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"$+&!\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k1G!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k2G$!$X%!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y 1G$\"$+&!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$yt1G$\"$3(!\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$estG$\"$3#!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG$\"$?\"!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 331 87 "el paso de h = 0.5 se acepta provisionalmente, y tomamos \+ y1 del m\351todo de orden 1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "verdadero_error:=yv(x1)-y1;error_admitido:=TU*(x1-xini);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%0verdadero_errorG$\"$n\"!\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%/error_admitidoG$\"$]#!\"$" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 333 23 "el nuevo paso es h = " } {XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT 335 49 "*h = 0.6, pero como \+ sobrepasa xend = 1 , h = 0.5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "h:=alpha*h;h:=1.-x1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG$ \"$+'!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG$\"$+&!\"$" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 334 7 "paso 2 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "x0:=x0+h;y0:=y1;x1:=x0+h;k1:=f(y0);k2:=f(y0+a[1,2 ]*h*k1);y1:=y0+h*(b[1]*k1+b[2]*k2);yt1:=y0+h*(bt[1]*k1+bt[2]*k2);est:= h*(d[1]*k1+d[2]*k2);alpha:=(TU*h)/abs(est);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G$\"$+&!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# y0G$\"$+&!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"$+\"!\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k1G$!$]#!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k2G$!$u\"!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# y1G$\"$v$!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$yt1G$\"$/%!\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$estG$\"$!H!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG$\"$i)!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 336 116 "el paso de h = 0.5 se acepta provisionalmente y hemos termi nado la integraci\363n; tomamos y1 del m\351todo de orden 1" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "verdadero_error:=yv(x1)-y1;e rror_admitido:=TU*(x1-xini);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0ver dadero_errorG$\"$D\"!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/error_a dmitidoG$\"$+&!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 338 43 "comportami ento en [0,1] de la soluci\363n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(yv(x),x=0..1);" }}{PARA 13 "" 1 "" {GLPLOT2D 496 496 496 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$\"\"!F)$\"\"\"F)7$$\"3emmm ;arz@!#>$\"3[gl2l#ymy*!#=7$$\"3[LL$e9ui2%F/$\"3*4Q1+\"yL3'*F27$$\"3nmm m\"z_\"4iF/$\"3Sq_(oR%Q:%*F27$$\"3[mmmT&phN)F/$\"3S*4]P$Q#)G#*F27$$\"3 CLLe*=)H\\5F2$\"3CP5qj&[.0*F27$$\"3gmm\"z/3uC\"F2$\"3S&HzHKP4*))F27$$ \"3%)***\\7LRDX\"F2$\"3LJwKyzoJ()F27$$\"3]mm\"zR'ok;F2$\"3!GQY(4P)Gd)F 27$$\"3w***\\i5`h(=F2$\"3=U*Qd*\\B?%)F27$$\"3WLLL3En$4#F2$\"3I#zmYw'yo #)F27$$\"3qmm;/RE&G#F2$\"3Qx%p:GL)R\")F27$$\"3\")*****\\K]4]#F2$\"3?1h BQ=R**zF27$$\"3$******\\PAvr#F2$\"3yzYAAo;jyF27$$\"3)******\\nHi#HF2$ \"3t.U-!*y?OxF27$$\"3jmm\"z*ev:JF2$\"3o;a'>fp(e&oF27$$\"3GLLeR\"3Gy%F2$\"3'>(pb:YhknF27$$\"3cmm;/T1&*\\F2 $\"3oNWD?6')omF27$$\"3&em;zRQb@&F2$\"3#p$>$>#)GAd'F27$$\"3\\***\\(=>Y2 aF2$\"31$R7Dlh.\\'F27$$\"39mm;zXu9cF2$\"3@8%[#zL?/kF27$$\"3l******\\y) )GeF2$\"3iR(*GRKcR &F27$$\"3=LLe9tOc()F2$\"3.!)H\"4!G_J`F27$$\"3u******\\Qk\\*)F2$\"3%yhc X.WrF&F27$$\"3CLL$3dg6<*F2$\"3$>!er(Hoh@&F27$$\"3ImmmmxGp$*F2$\"3U*>$[ [C\"G;&F27$$\"3A++D\"oK0e*F2$\"3.CvKdL62^F27$$\"3A++v=5s#y*F2$\"3c$p#y `j\"\\0&F27$F*$\"3++++++++]F2-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%+AX ESLABELSG6$%\"xGQ!6\"-%%VIEWG6$;F(F*%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}}{MARK "0 0 0" 6 } {VIEWOPTS 1 1 0 3 4 1802 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }