{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 278 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "Courier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 286 "Courier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 287 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 290 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 291 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 292 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 293 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 294 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 295 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 296 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 297 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 298 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 299 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 300 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 301 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 302 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 303 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 304 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 } {PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 269 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 270 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 271 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 272 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 273 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 273 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 277 1 " " }{TEXT 278 28 "An\341lisis II - Matem\341tica 3" }{TEXT 256 0 "" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 279 15 "TALLER DE MAPLE" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 14 " LABORATORIO 4" }}{PARA 259 "" 0 "" {TEXT -1 1 " " }} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 51 "Resoluci\363n de ecuaciones difer enciales: el comando " }{TEXT 280 6 "dsolve" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "res tart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Resolvamos la ecuaci\363 n diferencial:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "ec:=diff( y(x),x)+y(x)=sin(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dso lve(ec,y(x));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Pongamos la cond ici\363n y(0)=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "sol:=ds olve(\{ec,y(0)=0\},y(x));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Graf iquemos la soluci\363n hallada:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "y:=rhs(sol);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "plo t (y,x=-3..10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Otro ejemplo:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "ec:=diff(y(x),x,x)-2*diff(y(x),x)+10*y(x)=0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dsolve(ec,y(x));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 258 12 "Ejercicios:\n" }{TEXT -1 149 "a) Resolver la ecuac i\363n anterior con la condici\363n inicial y(0)=0.\n\nb) Resolver la \+ ecuaci\363n anterior con las condiciones iniciales y(0)=0 e D(y)(0)=1. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 " " 0 "" {TEXT -1 10 "El carrito" }{TEXT 257 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 317 "Una carreta de masa M est\341 sujeta a una pared por m edio de un resorte, que no ejerce fuerza cuando la carreta est\341 en \+ la posici\363n de equilibrio x=0. Si la carreta se desplaza a una dist ancia x, el resorte ejerce una fueza de restauraci\363n igual a -kx, d onde k es una constante positiva que mide la rigidez del resorte." }} {PARA 0 "" 0 "" {TEXT -1 57 "Por la segunda ley de movimiento de Newto n, se tiene que:" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "M*diff(x(t),t,t) \+ = -k*x(t);" "6#/*&%\"MG\"\"\"-%%diffG6%-%\"xG6#%\"tGF-F-F&,$*&%\"kGF&- F+6#F-F&!\"\"" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 12 "o, llama ndo " }{XPPEDIT 18 0 "a = sqrt(k/M);" "6#/%\"aG-%%sqrtG6#*&%\"kG\"\"\" %\"MG!\"\"" }{TEXT -1 11 ", se tiene:" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "diff(x(t),t,t)+a^2*x(t) = 0;" "6# /,&-%%diffG6%-%\"xG6#%\"tGF+F+\"\"\"*&%\"aG\"\"#-F)6#F+F,F,\"\"!" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 9 "Problema:" }{TEXT -1 104 " Si la carreta se lleva a la po sici\363n x(0)=x0 y se libera sin velocidad incial, hallar la funci \363n x(t).\n" }{TEXT 274 9 "Soluci\363n:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "ec:=diff(x(t),t,t)+a^2*x(t) = 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "sol:=dsolve(\{ec,x(0)=x0,D(x)(0)=0\},x(t));" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Grafiquemos la soluci\363n en el c aso M=1, k=0.5, x0=3:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " M:=1;\nk:=0.5;\nx0:=3;\na:=sqrt(k/M);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "x:=rhs(sol);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "plot(x,t=0..50);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 259 12 "Ejercicios: " }}{PARA 0 "" 0 "" {TEXT -1 100 "a) Cambie \+ x0. \277Cambia la frecuencia de oscilaci\363n? \277Y la amplitud?\nb) \+ \277Qu\351 sucede si cambia la masa?" }}{PARA 0 "" 0 "" {TEXT -1 43 "c ) \277Qu\351 sucede si el resorte es m\341s r\355gido?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "Si se produce una amortiguaci\363 n que se opone al movimiento, y de magnitud proporcional a la velocida d (= " }{XPPEDIT 18 0 "-c*diff(x(t),t);" "6#,$*&%\"cG\"\"\"-%%diffG6$- %\"xG6#%\"tGF-F&!\"\"" }{TEXT -1 122 ") debida al rozamiento, entonces la ecuaci\363n que describe el movimiento del carrito en funci\363n d el tiempo se convierte en:" }}{PARA 261 "" 0 "" {XPPEDIT 18 0 "M*diff( x(t),t,t) = -k*x(t)-c*diff(x(t),t);" "6#/*&%\"MG\"\"\"-%%diffG6%-%\"xG 6#%\"tGF-F-F&,&*&%\"kGF&-F+6#F-F&!\"\"*&%\"cGF&-F(6$-F+6#F-F-F&F3" } {TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 11 "o bien, si " }{XPPEDIT 18 0 "b = c/(2*M);" "6#/%\"bG*&%\"cG\"\"\"*&\"\"#F'%\"MGF'!\"\"" } {TEXT -1 5 " y " }{XPPEDIT 18 0 "a = sqrt(k/M);" "6#/%\"aG-%%sqrtG6# *&%\"kG\"\"\"%\"MG!\"\"" }{TEXT -1 11 ", se tiene:" }}{PARA 262 "" 0 " " {XPPEDIT 18 0 "diff(x(t),t,t)+2*b*diff(x(t),t)+a^2*x(t) = 0;" "6#/,( -%%diffG6%-%\"xG6#%\"tGF+F+\"\"\"*(\"\"#F,%\"bGF,-F&6$-F)6#F+F+F,F,*&% \"aGF.-F)6#F+F,F,\"\"!" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 260 29 "Movimiento sobreamortiguado: " }{TEXT -1 82 "la fuerza de rozamien to es grande en comparaci\363n con la rigidez del resorte (b>a)." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 18 "b:=5:\na:=2:\nx0:=3:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "ec:=diff(x(t),t, t)+2*b*diff(x(t),t)+a^2*x(t) = 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "sol:=dsolve(\{ec,x(0)=x0,D(x)(0)=0\},x(t));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "x:=rhs(sol);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "plot(x,t=0..15);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 261 11 "Ejercicio: " }{TEXT -1 23 "D\351 distintos valores a " }{TEXT 281 1 "a" }{TEXT -1 3 " y " }{TEXT 282 1 "b" }{TEXT -1 2 " (" } {TEXT 283 1 "b" }{TEXT -1 1 ">" }{TEXT 284 1 "a" }{TEXT -1 2 ")." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 26 "Movimiento subamortiguado:" }{TEXT 262 1 " " }{TEXT -1 81 "la fuerza de rozamiento es menor en comparaci\363n c on la rigidez del resorte (b " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "b:=2:\n a:=10:\nx0:=1:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "ec:=diff(x(t),t,t)+2*b*diff(x(t),t)+a^2*x (t) = 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "sol:=dsolve(\{e c,x(0)=x0,D(x)(0)=0\},x(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "x:=factor(rhs(sol));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "plot(x,t=0..3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 263 11 "Ejercicio: " }{TEXT -1 15 "\277Qu\351 sucede si " }{TEXT 285 1 "b" }{TEXT -1 16 " es cercano a 0?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 160 "Hasta ahora \+ hemos considerado vibraciones libres, porque solo actuan fuerzas inter nas al sistema. Si una fuerza F(t) act\372a sobre la carreta, la ecuac i\363n ser\341: " }}{PARA 265 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "M *diff(x(t),t,t)+c*diff(x(t),t)+k*x(t) = F(t);" "6#/,(*&%\"MG\"\"\"-%%d iffG6%-%\"xG6#%\"tGF.F.F'F'*&%\"cGF'-F)6$-F,6#F.F.F'F'*&%\"kGF'-F,6#F. F'F'-%\"FG6#F." }{TEXT -1 1 "." }}{PARA 264 "" 0 "" {TEXT -1 0 "" }} {PARA 263 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Consideremos la funci\363n " }{XPPEDIT 18 0 "F := PIECEWISE([0, t < 0 ],[20, -t < 0 and t-1/20 < 0],[0, 20 < t]);" "6#>%\"FG-%*PIECEWISEG6%7 $\"\"!2%\"tGF)7$\"#?32,$F+!\"\"F)2,&F+\"\"\"*&F4F4F-F1F1F)7$F)2F-F+" } {TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "M:=1;\nk:=1;\nc:=.5;\nF:=piecewise (t<0,0,t>0 and t<1,1,t>1,0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "ec:=M*diff(x(t),t,t)+c*diff(x(t),t)+k*x(t)=F;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "sol:=dsolve(\{ec,x(0)=0,D(x)(0)=0\},x(t)) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plot(rhs(sol),t=0..20) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "El siguiente ejemplo muestra una animaci\363n donde \+ la fuerza exterior es " }{XPPEDIT 18 0 "F := cos(w*t/40);" "6#>%\"FG -%$cosG6#*(%\"wG\"\"\"%\"tGF*\"#S!\"\"" }{TEXT -1 75 ", con w entre 1 \+ y 50. O sea la frecuencia impresa al sistema variar\341 entre " } {XPPEDIT 18 0 "1/40;" "6#*&\"\"\"F$\"#S!\"\"" }{TEXT -1 3 " y " } {XPPEDIT 18 0 "50/40;" "6#*&\"#]\"\"\"\"#S!\"\"" }{TEXT -1 232 " . Obs erve el fen\363meno de resonancia que se produce cuando la frecuencia \+ impresa es parecida a la frecuencia natural del sistema. En este ejemp lo la frecuencia natural es aproximadamente 0.99 y la frecuencia impre sa m\341s parecida es " }{XPPEDIT 18 0 "40/40 = 1" "6#/*&\"#S\"\"\"F%! \"\"F&" }{TEXT -1 43 " que se da en el cuadro 40 de la animaci\363n." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart;\nwith(plots):" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "M:=1;\nk:=1;\nc:=0.2;\nF:=cos(w/40 *t);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "for w from 1 to 50 do\nec:= M*diff(x(t),t,t)+c*diff(x(t),t)+k*x(t)=F;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "sol:=dsolve(\{ec,x(0)=0,D(x)(0)=0\},x(t));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "p[w]:=plot(rhs(sol),t=0..200);\nod:\n" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "display([seq(p[w],w=1..50)] ,insequence=true);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 264 10 "Ejercicio:" } {TEXT -1 80 " En la animaci\363n anterior considere el caso ideal dond e no hay rozamiento (c=0)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 48 "Sistemas de ecuaciones l ineales de primer orden." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "El comando " }{TEXT 286 6 "DEplot" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "restart:\nwith(DEtools):\n" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 15 "Sea el sistema:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "ec1 := diff(x(t),t) = -1/2*x(t)+y(t);" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 37 "ec2 := diff(y(t),t) = -x(t)-1/2*y(t);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Utilizaremos el comando " }{TEXT 275 6 "DEplot" }{TEXT -1 18 " para graficar el " }{TEXT 256 21 "campo \+ de direcciones:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "DEplot([ ec1,ec2], [x(t),y(t)], t=-10..10,x=-6..6,y=-6..6,\nthickness=1,scaling =constrained);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Tambi\351n pode mos graficar una o varias trayectorias especificando los " }{TEXT 276 15 "datos iniciales" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 289 "DEplot([ec1,ec2], [x(t),y(t)], t=-10..10, [[x(0)=1,y (0)=2],[x(0)=1,y(0)=4],[x(0)=-4,y(0)=1],[x(0)=-2,y(0)=1],[x(0)=1,y(0)= -2],[x(0)=1,y(0)=1],[x(0)=4,y(0)=5],\n [x(0)=1,y(0)=-4],[x(0)=2,y(0)= 1],[x(0)=4,y(0)=1]], x=-6..6, y=-6..6,stepsize=.1,thickness=1,linecolo r=blue,scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 "Forma matricial." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 241 "Escribiremos un sistema en forma \+ matricial y lo diagonalizaremos, o sea buscaremos un cambio de coorden adas que transforme el sistema en otro desacoplado. Luego compararemos cualitativamente los campos de direcciones y los diagramas de fase." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "restart;\nwith(DEtools): \nwith(linalg):\nwith(plots):\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "La matriz del sistema y el vector inc\363gnita son:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "A := matrix([[1,1],[4,1]]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "u:=[x(t),y(t)];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Hagamos el producto: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "M:=multiply(A,u);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "El sistema ser\355a:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "ec1:=diff(u[1],t)=M[1];\nec2:=diff(u[2],t)=M[2];\n" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Grafiquemos el campo de direccio nes y la curva de dato incicial u(0)=[1,-1]." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "DEplot([ec1,ec2],u,t=-2..2,[[x(0)=1,y(0)=-1]], \nx=-3..3,y=-3..3,stepsize=.1,thickness=1,linecolor=blue,scaling=const rained);\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "Diagonalicemos A. Para esto, primero busc amos los autovalores y autovectores de A con el comando " }{TEXT 265 12 "eigenvectors" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " [eigenvectors(A)];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 179 "Los autovalores son 3 (con multiplicidad 1) y -1 (con multipli cidad 1) y los respectivos autovectores que calcul\363 MAPLE son [1,2] y [1,-2].\nLa matriz de cambio de coordenadas es: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "C:=matrix([[1,1],[2,-2]]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Y su inversa:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 17 "invC:=inverse(C);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Para verificar hagamos el producto:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 25 "Diag:=multiply(invC,A,C);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "La idea \+ del cambio de coordenadas para desacoplar el sistema es la siguiente: \nPor ser " }{TEXT 287 4 "invC" }{TEXT -1 23 " inversible, el sistema " }}{PARA 266 "" 0 "" {TEXT -1 1 "\n" }{TEXT 266 6 "u'=A u" }}{PARA 0 "" 0 "" {TEXT -1 26 "es equivalente al sistema " }}{PARA 267 "" 0 "" {TEXT 267 16 "invC u'=invC A u" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Por ser" }{TEXT 288 5 " invC" }{TEXT -1 23 " constante res pecto de " }{TEXT 289 2 "t," }{TEXT -1 14 " se tiene que:" }}{PARA 268 "" 0 "" {TEXT -1 0 "" }{TEXT 268 18 "(invC u)'=invC A u" }}{PARA 0 "" 0 "" {TEXT -1 9 "Llamando " }{TEXT 290 8 "v=invC u" }{TEXT -1 17 " (y por lo tanto " }{TEXT 291 5 "u=C v" }{TEXT -1 2 "):" }}{PARA 269 "" 0 "" {TEXT -1 0 "" }{TEXT 269 13 "v'=invC A C v" }}{PARA 0 "" 0 "" {TEXT -1 5 "Pero " }{TEXT 292 13 "invC A C=Diag" }{TEXT -1 1 "," }} {PARA 270 "" 0 "" {TEXT -1 0 "" }{TEXT 270 9 "v'=Diag v" }}{PARA 0 "" 0 "" {TEXT -1 32 "\241y el sistema queda desacoplado!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "Luego basta hallar " } {TEXT 293 2 "v " }{TEXT -1 28 "(\241que es f\341cil!) y se tiene " } {TEXT 294 5 "u=C v" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 10 "Hag\341moslo:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "v:=[v1(t),v2(t)];\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "sistdesacoplado:=multiply(Diag,v);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "ecv1:=diff(v1(t),t)=sistdesacoplado[1];\n ecv2:=diff(v2(t),t)=sistdesacoplado[2];\n" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "\241Para resolver estas ecuaciones no ha ce falta usar MAPLE! (pero, si quiere puede hacerlo) Las soluciones so n:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "solv1:=c1*exp(3*t);\n solv2:=c2*exp(-t);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "donde c1 \+ y c2 son constantes." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Busquemos las soluciones del sistema original:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "multiply(C,[solv1,solv2]);" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 158 "Grafiquemos el campo de direcciones y la curva de dato incicial v(0)=invC u(0). Comp\341relo con el del sistema orginal y, s i quiere, agregue otras trayectorias." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "multiply(invC,[1,-1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "DEplot([ecv1,ecv2],v,t=-2..2,[[v1(0)=1/4,v2(0)=3/4]] ,v1=-3..3,v2=-3..3,stepsize=.1,thickness=1,linecolor=blue,scaling=cons trained);\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 271 11 "Ejercicio: " } {TEXT -1 308 "analice los sistemas lineales asociados a las siguientes matrices y observe su comportamiento seg\372n la parte real de sus au tovalores. Si quiere, d\351 ejemplos de los casos que faltan.\nNOTA: e n los dos \372ltimos ejemplos los autovalores no son reales; por lo ta nto no diagonalice los sistemas, s\363lo graf\355quelos.\n\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "A = matrix([[5, -2], [-1, 4]]) ;" "6#/%\"AG-%'matrixG6#7$7$\"\"&,$\"\"#!\"\"7$,$\"\"\"F-\"\"%" } {TEXT -1 9 " , " }{XPPEDIT 18 0 "A = matrix([[-3/2, 1/2], [1/2, \+ -3/2]]);" "6#/%\"AG-%'matrixG6#7$7$,$*&\"\"$\"\"\"\"\"#!\"\"F/*&F-F-F. F/7$*&F-F-F.F/,$*&F,F-F.F/F/" }{TEXT -1 8 " , " }{XPPEDIT 18 0 "A = matrix([[4, 5], [-5, 4]]);" "6#/%\"AG-%'matrixG6#7$7$\"\"%\"\"&7$,$ F+!\"\"F*" }{TEXT -1 5 " , " }{XPPEDIT 18 0 "A = matrix([[-1, 1], [- 1, -1]]);" "6#/%\"AG-%'matrixG6#7$7$,$\"\"\"!\"\"F+7$,$F+F,,$F+F," } {TEXT -1 3 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "Lineas de flujo de un campo gradi ente." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Dada un funci\363n diferenciab le f(x,y) se define el campo gradiente " }{XPPEDIT 18 0 "grad(f)(x,y) \+ = (diff(f(x,y),x), diff(f(x,y),y));" "6#/--%%gradG6#%\"fG6$%\"xG%\"yG6 $-%%diffG6$-F(6$F*F+F*-F.6$-F(6$F*F+F+" }{TEXT -1 189 ".\nEl gradiente en un punto se\361ala la direcci\363n de mayor crecimiento de la func i\363n y es opuesto a la de menor crecimiento. Adem\341s es perpendicu lar a la curva de nivel que pasa por ese punto." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "Se llaman curvas integral es de f(x,y) a las curvas [x(t),y(t)] que satisfacen el sistema:" }} {PARA 271 "" 0 "" {TEXT -1 5 "x'(t)" }{TEXT 272 1 "=" }{XPPEDIT 18 0 " diff(f(x(t),y(t)),x);" "6#-%%diffG6$-%\"fG6$-%\"xG6#%\"tG-%\"yG6#F,F* " }{TEXT -1 0 "" }}{PARA 272 "" 0 "" {TEXT -1 6 "y'(t)=" }{XPPEDIT 18 0 "diff(f(x(t), y(t)),y);" "6#-%%diffG6$-%\"fG6$-%\"xG6#%\"tG-%\"yG6#F ,F." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "Veamos un ejemplo: \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "restart;\nwith(DEtool s):\nwith(plots):\nwith(plottools):\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "\nLa funci\363n y su gr\341fica son:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 61 "f:=(x,y)->-5*x/(x^2 + y^2 + 1);\n\ngraf:=plo t3d(f,-3..3,-5..5):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "graf;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Grafiquemos el campo gradiente:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 205 "fx:=D[1](f):\nfy:=D[2](f) :\n\nec1:=diff(x(t),t)=fx(x,y):\nec2:=diff(y(t),t)=fy(x,y):\n\nfig1:=D Eplot([ec1,ec2], [x(t),y(t)], t=0..50, stepsize=.3, x=-3..3, y=-5..5, thickness=1,arrows=MEDIUM,color=black):\n\nfig1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 273 11 "Ejercicio: " }{TEXT -1 78 "Compare este gr\341fico con e l anterior. \277D\363nde est\341 el m\341ximo y d\363nde el m\355nimo? " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Ahora graficaremos algunas l\355neas de flujo:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 239 "fig2:=DEplot([ec1,ec2], [x( t),y(t)], t=0..30, [[x(0)=1,y(0)=2],[x(0)=2,y(0)=1],[x(0)=1,y(0)=-2],[ x(0)=1,y(0)=1]], stepsize=.1, x=-3..3,y=-5..5,thickness=3,\nlinecolor= [blue,green,red,yellow],arrows=NONE):\n\nfig3:=display([fig1,fig2]):\n \nfig3;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "Superpondremos este gr \341fico con el de la funci\363n (utilice el zoom para ver mejor):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Tcampo:=transform((x,y) -> \+ [x,y,0]):\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "display3d([graf,Tcam po(fig3)],style=patchnogrid);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 " Levantemos las l\355neas de flujo:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "Tf:=transform((x,y) -> [x,y,f(x,y)]):\n\ndisplay3d([g raf,Tcampo(fig1),Tf(fig2)],style=patchnogrid);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 201 "En los dos dibujos que siguen se agregan algunas cu rvas de nivel. Observe c\363mo son cortadas en forma perpendicular por el campo gradiente y las l\355neas de flujo. \nUtilice el zoom para \+ poder ver mejor.\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "disp lay3d([graf,Tcampo(fig3)],style=patchcontour,scaling=constrained);\ndi splay3d([graf,Tcampo(fig1),Tf(fig2)],style=patchcontour,scaling=constr ained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 "Ejercicios ecol\363gicos" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 41 "Modelo de especies que compiten entre s\355." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "En el siguiente ejercicio x(t) e \+ y(t) representan la cantidad de individuos de dos especies que compite n entre s\355." }}{PARA 0 "" 0 "" {TEXT -1 26 "El tiempo se mide en a \361os." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "with(DEtools):\nwith(plots): " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Las ecuaciones son:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "ec1:=diff(x(t),t)=3*x(t)-y(t );\nec2:=diff(y(t),t)=-2*x(t)+2*y(t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Si y(0)>2x(0) entonces la especie " }{TEXT 295 1 "x" } {TEXT -1 33 " resulta eliminada con el tiempo." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 37 "linea:=plot(2*x,x=0..400,color=blue):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "campo:=DEplot([ec1,ec2],[x(t),y(t) ],t=0..1,[[x(0)=90,y(0)=150],[x(0)=90,y(0)=200]],stepsize=.2,\nx=0..70 0,y=0..800,thickness=1,color=green,linecolor=red):\ndisplay(linea,camp o);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Si la poblaci\363n inicial es x(0)=90 e y(0)=150 entonces la especie " }{TEXT 296 2 "y " }{TEXT -1 28 "desaparece luego de 11 meses" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "sol:=dsolve( \{ec1,ec2,x(0)=90,y(0)=150\},\{y(t),x(t) \});" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 41 "fsolve(-10*exp(4*t)+160*exp(t)=0,t=0..1);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf (11/12);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "Modelo predador-presa." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Consideremos el siguiente modelo d onde la especie " }{TEXT 297 1 "x" }{TEXT -1 18 " es la presa y la " } {TEXT 298 1 "y" }{TEXT -1 111 " el predador.\n\nProbaremos que cualqui era sean las poblaciones iniciales la presa se extinguir\341 antes de \+ un a\361o." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 14 "with(DEtools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "ec1:=diff(x(t),t)=2*x(t)-y(t);\nec2:=diff(y(t),t)=x(t )+4*y(t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Llamaremos " }{TEXT 299 1 "a" }{TEXT -1 3 " y " }{TEXT 300 1 "b" }{TEXT -1 32 " a las pobl aciones iniciales de " }{TEXT 301 1 "x" }{TEXT -1 3 " e " }{TEXT 302 1 "y" }{TEXT -1 17 " respectivamente." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "sol:=dsolve( \{ec1,ec2,x(0)=a,y(0)=b\},\{y(t),x(t)\}) ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "La especie " }{TEXT 303 1 "x" }{TEXT -1 17 " se extingir \341 en:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 22 "solve(-a+t*a+t*b=0,t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Que es menos que un a\361o.\n\nPor ejemplo si a=500 \+ y b=100 entonces " }{TEXT 304 1 "x" }{TEXT -1 24 " desaparece en 10 me ses:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "500/(500+100)=evalf (500/(500+100));\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "DEplot([ec1, ec2],[x(t),y(t)],t=0..0.83333,[[x(0)=500,y(0)=100]],stepsize=.2,\nx=0. .1000,y=0..8000,thickness=1,color=green,linecolor=red);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 "Modelo de cooperaci\363n de especies." }}{PARA 4 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 240 "El siguiente sistema d e ecuaciones modela un sistema de dos especies que cooperan entre s \355.\nVeremos que cualquiera sean las poblaciones iniciales (x(0)=a e y(0)=b positivos) ambas poblaciones alcanzan un equilibrio, sin desap arecer ninguna." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "with(DEtools):\nwith(pl ots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "ec1:=diff(x(t),t)= -2*x(t)+4*y(t);\nec2:=diff(y(t),t)=x(t)-2*y(t);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 50 "sol:=dsolve( \{ec1,ec2,x(0)=a,y(0)=b\},\{x(t), y(t)\});" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Veamos que pasa en el infinito:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 31 "limit(rhs(sol[2]),t=infinity);\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "limit(rhs(sol[1]),t=infinity);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 125 "Se observa que ninguna desaparece. \nPor ejemplo si a=100 y b=300 entonces, con el tiempo, las poblaciones se \+ equilibraran en:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "300+100 /2;\n300/2+100/4;\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Grafiquemo s este caso junto con la recta de equilibrio (en azul):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "campo:=DEplot([ec1,ec2],[x(t),y(t) ],t=0..100,[[x(0)=100,y(0)=300]],stepsize=.2,x=0..400,y=0..300,thickne ss=1,color=green,linecolor=red):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "equilibrio:=implicitplot(-2*x+4*y=0,x=0..400,y=0..400,color=blue):\n " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "display(campo,equilibrio);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}}}{MARK "1 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }