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{TEXT 268 14 "Ejercicios de " }}{PARA 4 "" 0 "" {TEXT 288 22 "(8) Estabilidad lineal" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 287 52 "Ejercicio 08-20 (del E XAMEN ORDINARIO de 01FEB06) " }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 83 "a) B\372squense los m\351todos con el mayor ord en posible para la familia de RUNGE-KUTTA " }}{PARA 4 "" 0 "" {TEXT -1 32 "impl\355citos del siguiente tablero" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 256 18 " 0 | 0" }}{PARA 4 "" 0 "" {TEXT 257 5 " " }{XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT 264 9 " | " }{XPPEDIT 18 0 "1-alpha;" "6#,&\"\"\"F$%&alphaG!\"\" " }{TEXT 275 8 " " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }} {PARA 4 "" 0 "" {TEXT -1 12 " ---------" }{TEXT 258 24 "------------ ------------" }}{PARA 4 "" 0 "" {TEXT 259 20 " | " } {XPPEDIT 18 0 "b[1]" "6#&%\"bG6#\"\"\"" }{TEXT 265 10 " " } {XPPEDIT 18 0 "b[2]" "6#&%\"bG6#\"\"#" }{TEXT 266 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 72 "b) Para los m\351to dos de la familia, calc\372lese la funci\363n de estabilidad." }} {PARA 4 "" 0 "" {TEXT -1 59 "H\341gase lo mismo para los m\351todos de orden m\341ximo obtenidos." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 58 "c) Disc\372tase si estos \372ltimos m\351todos \+ son o no A-estables." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "rest art:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "interface(labeling= false):" }}}{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 37 "y3:=subs(di ff(y(x),x)=y1,diff(y2,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 28 "Y3:=subs(\{y(x)=y0,x=x0\},y3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "k1_:=unapply(f(x0,y0),h):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "k2_:=unapply(f(x0+beta*h,y0+ (1-alpha)*h*kp1_(h)+alpha*h*kp2_(h)),h):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "k1_0:=k1_(0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "k2_0:=k2_(0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "k1 1:=D(k1_):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "k21:=D(k2_): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "k110:=subs(\{kp1_(0)=k1 _0,kp2_(0)=k2_0\},k11(0)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "k210:=subs(\{kp1_(0)=k1_0,kp2_(0)=k2_0\},k21(0)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "k12:=D(k11):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 12 "k22:=D(k21):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "k120:=subs(\{D(kp1_)(0)=k110,D(kp2_)(0)=k210,kp1_(0)= k1_0,kp2_(0)=k2_0\},k12(0)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "k220:=subs(\{D(kp1_)(0)=k110,D(kp2_)(0)=k210,kp1_(0)=k1_0,kp2_(0 )=k2_0\},k22(0)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 263 2 "a)" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 260 27 "condiciones para el orden 1" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Fi_0:=expand(b[1]*k1_0+b[2]* k2_0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Fi_0G,&*&&%\"bG6#\"\"\"F* -%\"fG6$%#x0G%#y0GF*F**&&F(6#\"\"#F*F+F*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/\"\"\",&&%\"bG6#F&F&&F)6# \"\"#F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "solve(cond1a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/&%\"bG6#\"\"#,&\"\"\"F*&F&6#F*!\" \"/F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 261 27 "condiciones para el o rden 2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Fi10:=expand(b[1] *k110+b[2]*k210);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Fi10G,&*(&%\"b G6#\"\"#\"\"\"--&%\"DG6#F+6#%\"fG6$%#x0G%#y0GF+%%betaGF+F+*(F'F+--&F/F )F1F3F+-F2F3F+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/#\"\"\"\"\"#*&&%\"bG6#F(F'%%betaG F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "cond2b:=(1/2)*coeff(c oeff(Y2,D[2](f)(x0,y0)),f(x0,y0))=coeff(coeff(Fi10,D[2](f)(x0,y0)),f(x 0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond2bG/#\"\"\"\"\"#&%\" bG6#F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "solucion1:=solve( \{cond1a,cond2a,cond2b\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*soluc ion1G<%/%%betaG\"\"\"/&%\"bG6#F(#F(\"\"#/&F+6#F.F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 28 "condiciones para el orden 3 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Fi20:=expand(b[1]*k120+b[2]*k220);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%%Fi20G,,*(&%\"bG6#\"\"#\"\"\"--&%\"D G6$F+F+6#%\"fG6$%#x0G%#y0GF+)%%betaGF*F+F+*,F*F+F'F+F7F+--&F/6$F+F*F1F 3F+-F2F3F+F+*(F'F+--&F/6$F*F*F1F3F+)F=F*F+F+*.F*F+F'F+--&F/F)F1F3F+%&a lphaGF+--&F/6#F+F1F3F+F7F+F+*,F*F+F'F+)FEF*F+FHF+F=F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "cond3a:=(1/6)*coeff(Y3,D[1,1](f)(x0 ,y0))=(1/2)*coeff(Fi20,D[1,1](f)(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3aG/#\"\"\"\"\"',$*&#F'\"\"#F'*&&%\"bG6#F,F')%%b etaGF,F'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "cond3b:=(1 /6)*coeff(coeff(Y3,D[1,2](f)(x0,y0)),f(x0,y0))=(1/2)*coeff(coeff(Fi20, D[1,2](f)(x0,y0)),f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'con d3bG/#\"\"\"\"\"$*&&%\"bG6#\"\"#F'%%betaGF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "cond3c:=(1/6)*coeff(coeff(Y3,D[2,2](f)(x0,y0)),f( x0,y0)^2)=(1/2)*coeff(coeff(Fi20,D[2,2](f)(x0,y0)),f(x0,y0)^2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3cG/#\"\"\"\"\"',$*&#F'\"\"#F'& %\"bG6#F,F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "cond3d:=( 1/6)*coeff(coeff(Y3,D[2](f)(x0,y0)),D[1](f)(x0,y0))=(1/2)*coeff(coeff( Fi20,D[2](f)(x0,y0)),D[1](f)(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3dG/#\"\"\"\"\"'*(&%\"bG6#\"\"#F'%&alphaGF'%%betaGF'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "cond3e:=(1/6)*coeff(coeff(Y 3,D[2](f)(x0,y0)^2),f(x0,y0))=(1/2)*coeff(coeff(Fi20,D[2](f)(x0,y0)^2) ,f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3eG/#\"\"\"\"\"' *&&%\"bG6#\"\"#F'%&alphaGF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "solve(\{cond1a,cond2a,cond2b,cond3a,cond3b,cond3c,cond3d,cond3e\}) ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 269 69 "Es decir, no hay m\351todos de orden 3 . Existe una familia infinita de" }}{PARA 0 "" 0 "" {TEXT 295 54 "m\351todos de orden 2 , la que viene dada por el tabler o" }}{PARA 4 "" 0 "" {TEXT 289 18 " 0 | 0" }}{PARA 4 "" 0 "" {TEXT 290 16 " 1 | " }{XPPEDIT 18 0 "1-alpha;" "6#,&\"\" \"F$%&alphaG!\"\"" }{TEXT 294 8 " " }{XPPEDIT 18 0 "alpha;" "6# %&alphaG" }}{PARA 4 "" 0 "" {TEXT -1 12 " ---------" }{TEXT 291 24 " ------------------------" }}{PARA 4 "" 0 "" {TEXT 292 20 " | " }{XPPEDIT 18 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 293 9 " " }{XPPEDIT 18 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }}{PARA 0 " " 0 "" {TEXT 296 24 "uno de cuyos m\351todos es " }{TEXT 279 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "solucion2:=subs(alpha=1/2,so lucion1) union \{alpha=1/2\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*so lucion2G<&/%%betaG\"\"\"/&%\"bG6#F(#F(\"\"#/&F+6#F.F-/%&alphaGF-" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 286 46 "el conocido como LOBATTO IIIA \+ de 2 etapas" }}{PARA 4 "" 0 "" {TEXT 280 18 " 0 | 0" }} {PARA 4 "" 0 "" {TEXT 281 16 " 1 | " }{XPPEDIT 18 0 "1/2" " 6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 285 9 " " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }}{PARA 4 "" 0 "" {TEXT -1 12 " ---------" }{TEXT 282 24 "------------------------" }}{PARA 4 "" 0 "" {TEXT 283 17 " | " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" } {TEXT 284 10 " " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\" " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 270 2 "b)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 267 23 "funci\363n de estabilid ad " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f:=(x,y)->lambda*y; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"yG6\"6$%)operato rG%&arrowGF)*&%'lambdaG\"\"\"9%F/F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "ka1:=unapply(f(x0,y0),h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ka1Gf*6#%\"hG6\"6$%)operatorG%&arrowGF(*&%'lambdaG\" \"\"%#y0GF.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "ka2:=u napply(f(x0+beta*h,y0+(1-alpha)*h*kp1(h)+alpha*h*kp2(h)),h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ka2Gf*6#%\"hG6\"6$%)operatorG%&arrowGF(*& %'lambdaG\"\"\",(%#y0GF.*(,&F.F.%&alphaG!\"\"F.9$F.-%$kp1G6#F5F.F.*(F3 F.F5F.-%$kp2GF8F.F.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "test:=solve(\{k1=subs(\{kp1(h)=k1,kp2(h)=k2\},ka1(h)),k2=subs(\{kp 1(h)=k1,kp2(h)=k2\},ka2(h))\},\{k1,k2\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%testG<$/%#k1G*&%'lambdaG\"\"\"%#y0GF*/%#k2G**F)F*F+F *,(F*!\"\"*&F)F*%\"hGF*F0*(F)F*%&alphaGF*F2F*F*F*,&F*F0F3F*F0" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "k1:=subs(test,k1);k2:=subs(t est,k2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k1G*&%'lambdaG\"\"\"%#y 0GF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k2G**%'lambdaG\"\"\"%#y0GF' ,(F'!\"\"*&F&F'%\"hGF'F**(F&F'%&alphaGF'F,F'F'F',&F'F*F-F'F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "y1:=expand(y0+h*(b[1]*k1+b[2 ]*k2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G,,%#y0G\"\"\"**%\"hGF '&%\"bG6#F'F'%'lambdaGF'F&F'F'*,F)F'&F+6#\"\"#F'F-F'F&F',&F'!\"\"*(F-F '%&alphaGF'F)F'F'F3F3*,F/F'F-F1F&F'F2F3F)F1F3*.F/F'F-F1F&F'F2F3F5F'F)F 1F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "y1:=collect(subs(lam bda=z/h,y1),y0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G*&,,\"\"\"F' *&&%\"bG6#F'F'%\"zGF'F'*(&F*6#\"\"#F'F,F',&F'!\"\"*&F,F'%&alphaGF'F'F2 F2*(F.F'F,F0F1F2F2**F.F'F,F0F1F2F4F'F'F'%#y0GF'" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 41 "r:=unapply(collect(simplify(y1/y0),z),z);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rGf*6#%\"zG6\"6$%)operatorG%&arrow GF(*&,(\"\"\"!\"\"*&,(*&&%\"bG6#F.F.%&alphaGF.F.&F46#\"\"#F/*&F7F.F6F. F.F.)9$F9F.F.*&,(F6F.F3F/F7F/F.F " 0 "" {MPLTEXT 1 0 46 "r1:=unapply( simplify(subs(solucion1,r(z))),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#r1Gf*6#%\"zG6\"6$%)operatorG%&arrowGF(,$*&#\"\"\"\"\"#F/*&,,F0!\"\" *(F0F/)9$F0F/%&alphaGF/F/*$F5F/F3*(F0F/F6F/F7F/F/*&F0F/F6F/F3F/,&F/F3* &F6F/F7F/F/F3F/F/F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 272 49 "lo es de las soluciones de orden 2 obtenidas, y" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "r2:=unapply(simplify(subs(solucion2,r(z))),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r2Gf*6#%\"zG6\"6$%)operatorG%&arr owGF(,$*&,&\"\"#\"\"\"9$F0F0,&F/!\"\"F1F0F3F3F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 298 53 "lo es del citado m\351todo LO BATTO IIIA de 2 etapas" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 313 2 "c)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 312 55 "funci\363n de estabilidad para los m\351todos del mayor orden" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 297 92 "Para la familia d e los m\351todos de orden 2 obtenidos, el dominio de estabilidad, co mpuesto " }}{PARA 0 "" 0 "" {TEXT 277 13 "por los z = " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT 273 80 " h tales que el m\363dulo de r1(z) sea inferior a 1 no ser\341 en general todo el" }}{PARA 0 " " 0 "" {TEXT 274 84 "semiplano de parte real negativa, ya que la funci \363n de estabilidad tiende a infinito" }}{PARA 0 "" 0 "" {TEXT 278 92 "cuando z tiende a infinito en cualquier direcci\363n. O sea, los m\351todos no ser\341n A-estables" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT 299 17 "Claro que, si 2 " }{XPPEDIT 18 0 "alph a;" "6#%&alphaG" }{TEXT 300 18 " - 1 = 0 , o sea " }{XPPEDIT 18 0 "al pha;" "6#%&alphaG" }{TEXT 301 47 " = 1/2 , numerador y denominador son polinomios" }}{PARA 0 "" 0 "" {TEXT 302 81 "de igual grado y la cosa \+ cambia. Porque, en ese caso, el m\351todo es justamente el " }}{PARA 0 "" 0 "" {TEXT 303 9 "ya citado" }{TEXT 304 79 " LOBATTO IIIA de \+ 2 etapas, cuya funci\363n de estabilidad ya se ha exhibido. " }} {PARA 0 "" 0 "" {TEXT 305 11 "En ese caso" }{TEXT -1 3 " , " }{TEXT 306 50 "el dominio de estabilidad, compuesto por los z = " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT 307 25 " h tales que el m\363dulo " }}{PARA 0 "" 0 "" {TEXT 308 86 "de r2(z) sea inferior a 1 , esta r\341 formado por los valores de z que disten menos " }}{PARA 0 "" 0 "" {TEXT 309 87 "del punto -2 que del punto 2 . Esto corresponde \+ al semiplano de parte real negativa," }}{PARA 0 "" 0 "" {TEXT 310 83 " por lo que el m\351todo LOBATTO IIIA de 2 etapas es A-estable , y es el \372nico " }}{PARA 0 "" 0 "" {TEXT 311 51 "que lo es de entr e los de orden 2 , m\341ximo posible." }{TEXT -1 0 "" }}}}{MARK "0 3 0 " 15 }{VIEWOPTS 1 1 0 3 4 1802 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }