{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "MS Serif" 1 12 0 128 128 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "2D Output" -1 20 "Times" 1 12 0 0 255 1 2 2 2 2 2 1 0 0 0 1 }{CSTYLE "Text" -1 200 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 206 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 207 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 } {CSTYLE "" -1 208 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE " " -1 220 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 230 " Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 241 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 246 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times " 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 } {PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "MS Serif" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 2 0 2 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "MS Serif" 1 16 0 0 0 1 2 1 2 2 2 2 1 1 1 1 } 1 1 0 0 8 2 2 0 2 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 1 0 0 0 0 2 0 2 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 2 0 2 0 2 2 0 1 }{PSTYLE "No Na me211" -1 200 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "No Name225" -1 204 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "No Name212" -1 205 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 } {PSTYLE "No Name224" -1 207 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "No Name222" -1 209 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "No Name208" -1 211 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 2 0 2 0 2 2 0 1 }{PSTYLE "No Name221" -1 212 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "No Na me210" -1 213 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "No Name209" -1 215 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "No Name223" -1 216 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 } {PSTYLE "No Name" -1 218 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "No Name" -1 219 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "No Name" -1 220 1 {CSTYLE "" -1 -1 " MS Serif" 1 16 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "" 218 256 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 219 257 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 200 48 "MAT1154 - Equa\347\365e s Diferenciais e de Diferen\347as " }}{PARA 257 "" 0 "" {TEXT 200 60 " Retrato de Fase de Sistemas Lineares de Ordem 1 e Dimens\343o 2" }} {PARA 211 "" 0 "" {TEXT -1 0 "" }}{PARA 215 "" 0 "" {TEXT 206 16 "Ackn owledgement:" }{TEXT 200 71 " Este script foi baseado em um semelhante obtido no site da MapleSoft. " }}{PARA 213 "" 0 "" {TEXT 207 12 "Reco menda\347\343o" }{TEXT 200 93 " : Ao aluno \351 fortemente recome ndado que ele navegue no Help do Maple e veja informa\347\365es" }} {PARA 200 "" 0 "" {TEXT 200 118 " mais detalhadas sobre os comandos utilizados. Ningu\351m precisa de tantos comandos, mas" }}{PARA 0 "" 0 "" {TEXT -1 79 " \+ cada um ddeles poupa um pouquinho de trabalho." }}{PARA 205 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with( DEtools ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with( plots ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with( linalg ):" }}}{SECT 0 {PARA 3 "" 0 "13.A-1" {TEXT -1 6 "Caso 1" }}{EXCHG {PARA 0 "" 0 "" {TEXT 200 57 "Considere o seguinte sistem a aut\364nomo de EDO's lineares:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "sys := [ diff( x(t), t ) = x(t) - y(t) ," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 " diff( y(t), t ) = 2*x(t) +4*y(t) ];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$sysG7$/-%%diffG6$-%\"xG6#%\"tGF-,&F *\"\"\"-%\"yGF,!\"\"/-F(6$F0F-,&*&\"\"#F/F*F/F/*&\"\"%F/F0F/F/" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 200 108 "Se colocado numa forma matricial (comando meio firula), o sistema acima possui a seguinte matriz de si stema:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "A := genmatrix( map(rhs,s ys), [x(t),y(t)], `-b` );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AGK%' matrixG6#7$7$\"\"\"!\"\"7$\"\"#\"\"%Q(pprint06\"" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 200 64 "Seus pares de autovalores e autovetores podem ser calculados via" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "z:= eigenvectors (A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG6$7%\"\"$\"\"\"<#K%'vect orG6#7$F(!\"#Q(pprint06\"7%\"\"#F(<#KF+6#7$!\"\"F(Q(pprint1F0" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 200 39 "Assim estamos diante de um sistem a com " }{TEXT 208 45 "autovalores reais, de mesmo sinal e distintos" }{TEXT 200 2 ". " }}{PARA 0 "" 0 "" {TEXT 200 74 "A origem \351 o \372 nico equil\355brio, um repulsor (os autovalores s\343o positivos)." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "lambda1:=abs(z[1][1]);# A primeira coordenada de z, que \351 z[1], \351 a tripla \{2,1,\{[-1,1]\}\}.\nla mbda2:=abs(z[2][1]);\nv1:=z[1][3][1]; # A posi\347\343o z[1][3] \351 o autovetor associado a 2.\nv2:=z[2][3][1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(lambda1G\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%( lambda2G\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v1GK%'vectorG6#7$ \"\"\"!\"#Q(pprint06\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v2GK%'vec torG6#7$!\"\"\"\"\"Q(pprint06\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 19 "Atrav\351s do comando " }{HYPERLNK 17 "DEplot" 2 "DEplot" "" } {TEXT 200 70 " podemos criar o campo de dire\347\365es deste sistema e atrav\351s do comando " }{HYPERLNK 17 "pointplot" 2 "pointplot" "" } {TEXT 200 199 " podemos marcar alguns pontos convenientes.\nOs pontos \+ verde e azul, que indicam as retas pela origem contendo os autovetores . Se tra\347\341ssemos as duas retas, elas passariam por vetores tange ntes a elas" }}{PARA 0 "" 0 "" {TEXT 200 149 "em toda sua extens\343o. Ali\341s, cada reta s\343o tr\352s trajet\363rias: a origem isoladame nte \351 uma (um equil\355brio) e as outras duas semi-retas sem a orig em s\343o" }}{PARA 0 "" 0 "" {TEXT 200 11 "mais duas. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 387 "dir_field1 := DEplot( sys, \{x(t),y(t)\}, t=0..1, x=-3..3, y=-3..3, scene=[x,y] ): \nequil_plot1a := pointplot( v1, symbol=CIRCLE, symbolsize=12, color=B LUE ):\nequil_plot1b := pointplot( v2, symbol=CIRCLE, symbolsize=12, c olor=GREEN ):\nequil_plot1c := pointplot( [0,0], symbol=CIRCLE, symbol size=12, color=BLACK ):\ndisplay( [dir_field1,equil_plot1a,equil_plot1 b, equil_plot1c], axes=BOXED );" }}{PARA 13 "" 1 "" {GLPLOT2D 613 613 613 {PLOTDATA 2 "6)-%'CURVESG6^dlX4%)anythingG6\"6\"[gl'!%\"!!#'\"$\"# !!!!!!!!C008000000000000C006DCF7EA712DD0C008000000000000C0092308158ED2 30C007B73DFA9C4B74C0086ACE0D74BB00X4F'F(F)[gl'!%\"!!#'\"$\"#!!!!!!!!C0 07FA8283413A02C00456488ACC4062C008057D7CBEC5FEC0069C3E31BFBB5AC007B945 16E703ECC005E56BED54A096X4F'F(F)[gl'!%\"!!#'\"$\"#!!!!!!!!C007F4218C02 4222C001CFBCA2E25C9FC0080BDE73FDBDDEC0041550D67A5307C007BBA88C6A5C57C0 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No te que cada escolha [x(0)=...,y(0)=...] corresponde a seis pontos, pel a varia\347\343o do i." }}{PARA 0 "" 0 "" {TEXT -1 295 "Ao todo dever \355amos ver 24 \363rbitas (trajet\363rias) na figura, mas muitos dele s est\343o fora da janela escolhida. \nVamos desenhar as retas pelos a utovetores [-1,1] e [1,-2]. ligando dois pontos extremos na janela.\nN ote que n\343o se v\352 meia reta (em preto) associada ao autovetor [1 ,-2]: explique porque." }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{MPLTEXT 1 0 557 "IC :=[ [x(0)= i,y(0)= 2*i] $ i=0..5, # [1,2] est\341 na janela , [2,4] n\343o est\341.\n [x(0)=-i,y(0)= -3*i] $ i=0..5, \n \+ [x(0)=-i,y(0)= 3-i] $ i=0..5,\n [x(0)= i,y(0)= i-7] $ i=0..5 ]:\nautovet1 := pointplot( \{[-3,3],[3,-3]\}, connect=true, thickness= 3, color=BLACK ):\nautovet2 := pointplot( \{[1.5,-3],[-1.5,3]\}, conne ct=true, thickness=3, color=BLACK ):\nphase_port1 := DEplot( sys, \{x( t),y(t)\}, t=-3..3,x=-3..3, y=-3..3, IC, scene=[x,y],arrows=NONE, line color=CYAN, stepsize=0.25 ):\ndisplay([dir_field1,phase_port1,autovet1 , autovet2], axes=BOXED);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 613 613 613 {PLOTDATA 2 "6?-%'CURVESG6^dlX4%)anythingG6\"6\"[gl'!%\"!!#'\"$\"#!!!! !!!!C008000000000000C006DCF7EA712DD0C008000000000000C0092308158ED230C0 07B73DFA9C4B74C0086ACE0D74BB00X4F'F(F)[gl'!%\"!!#'\"$\"#!!!!!!!!C007FA 8283413A02C00456488ACC4062C008057D7CBEC5FEC0069C3E31BFBB5AC007B94516E7 03ECC005E56BED54A096X4F'F(F)[gl'!%\"!!#'\"$\"#!!!!!!!!C007F4218C024222 C001CFBCA2E25C9FC0080BDE73FDBDDEC0041550D67A5307C007BBA88C6A5C57C00360 35A7DE3738X4F'F(F)[gl'!%\"!!#'\"$\"#!!!!!!!!C007ECA40FE1C463BFFE92CE12 5CFF6EC008135BF01E3B9DC0018E2D2CBA2BA9C007BE81FFB9995DC000DB329581E3A8 X4F'F(F)[gl'!%\"!!#'\"$\"#!!!!!!!!C007E3BDA7F4FCD3BFF986CAE595B6A2C008 1C42580B032DBFFE0D6AFF539796C007C1F4E7F276E8BFFCACD5D25EEE10X4F'F(F)[g l'!%\"!!#'\"$\"#!!!!!!!!C007D90691DE9CB2BFF47BCFB7B9A38DC00826F96E2163 4EBFF8FD73A6D11281C007C63358229BC6BFF7A3CDEF1A30B2X4F'F(F)[gl'!%\"!!#' 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10 0 2 9 1 2 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Cur ve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 1 8" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" " Curve 25" "Curve 26" }}}}}{SECT 0 {PARA 3 "" 0 "13.A-1" {TEXT -1 6 "Ca so 2" }}{EXCHG {PARA 0 "" 0 "" {TEXT 200 16 "Agora considere " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "sys := [ diff( x(t), t ) = x(t) \+ - y(t) ," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 " diff( y(t), \+ t ) = 2*x(t) -4*y(t) ];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$sysG7$/ -%%diffG6$-%\"xG6#%\"tGF-,&F*\"\"\"-%\"yGF,!\"\"/-F(6$F0F-,&*&\"\"#F/F *F/F/*&\"\"%F/F0F/F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 20 "com matr iz associada" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "A := genmatrix( ma p(rhs,sys), [x(t),y(t)], `-b` );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"AGK%'matrixG6#7$7$\"\"\"!\"\"7$\"\"#!\"%Q(pprint06\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 28 "e autovalores e autovetores " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "z:= eigenve ctors(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG6$7%,&#\"\"$\"\"#! \"\"*&F*F+\"#<#\"\"\"F*F/F/<#K%'vectorG6#7$,&#\"\"&\"\"%F/*&F8F+F-F.F/ F/Q(pprint06\"7%,&#F)F*F+*&F*F+F-F.F+F/<#KF26#7$,&F6F/*&F8F+F-F.F+F/Q( pprint1F;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 39 "Assim estamos diant e de um sistema com " }{TEXT 220 38 "autovalores reais de sinais difer entes" }{TEXT 200 50 ". O \372nico equil\355brio, a origem, agora \351 uma sela. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "lambda1:=evalf(z[1][ 1]);\nv1:=evalf(z[1][3][1]);\nlambda2:=evalf(z[2][1]);\nv2:=evalf(z[2] [3][1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(lambda1G$\"*8Gbh&!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v1GK%'vectorG6#7$$\"+1kx!G#!\"*$ \"\"\"\"\"!Q(pprint06\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(lambda2G $!+8GbhN!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v2GK%'vectorG6#7$$ \"*%fB#>#!\"*$\"\"\"\"\"!Q(pprint06\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 19 "Atrav\351s do comando " }{HYPERLNK 17 "DEplot" 2 "DEplot " "" }{TEXT 200 70 " podemos criar o campo de dire\347\365es deste sis tema e Atrav\351s do comando " }{HYPERLNK 17 "pointplot" 2 "pointplot " "" }{TEXT 200 8 " podemos" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 207 "dir _field1 := DEplot( sys, \{x(t),y(t)\}, t=0..1, x=-3..3, y=-3..3, scene =[x,y] ):\nequil_plot1 := pointplot( [0,0], symbol=CIRCLE, symbolsize= 12, color=BLACK ):\ndisplay( [dir_field1,equil_plot1], axes=BOXED );" }}{PARA 13 "" 1 "" {GLPLOT2D 613 613 613 {PLOTDATA 2 "6'-%'CURVESG6^dl X4%)anythingG6\"6\"[gl'!%\"!!#'\"$\"#!!!!!!!!C008000000000000C00923081 58ED230C008000000000000C006DCF7EA712DD0C00848C20563B48CC0079531F28B450 0X4F'F(F)[gl'!%\"!!#'\"$\"#!!!!!!!!C007ECA40FE155B5C0069BA6700BCC76C00 8135BF01EAA4BC00456E04C802F46C0084FB3889CAB20C0051388DBC7AC15X4F'F(F)[ gl'!%\"!!#'\"$\"#!!!!!!!!C007CBF04DE411BAC00410DD104CCD74C008340FB21BE E46C001D430690FE232C0085AB0AD3CC16EC0029675A35F4685X4F'F(F)[gl'!%\"!!# '\"$\"#!!!!!!!!C0078D5B6745E4D1C001774A29FC97B1C00872A498BA1B2FBFFEC09 417D8275EC0086CF293AC285FC0002647F155BA8BX4F'F(F)[gl'!%\"!!#'\"$\"#!!! 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*%fB#>#F[il$\"#5FgdlFeauFgau-%*AXESSTYLEG6#%$BOXG-%+AXESLABELSG6%Q\"xF )Q\"yF)-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$!+PZ*y:$Faau$\"+PZ*y:$FaauFfcu " 1 2 0 1 10 0 2 9 1 2 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "C urve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "C urve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve \+ 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "Cur ve 28" "Curve 29" }}}}}{SECT 0 {PARA 3 "" 0 "13.A-1" {TEXT -1 6 "Caso \+ 3" }}{EXCHG {PARA 0 "" 0 "" {TEXT 200 10 "Considere " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "sys2 := [ diff( x(t), t ) = 3*x(t) - y(t) ," } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 " diff( y(t), t ) = 4*x(t) + 7*y(t) ];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sys2G7$/-%%diffG6 $-%\"xG6#%\"tGF-,&*&\"\"$\"\"\"F*F1F1-%\"yGF,!\"\"/-F(6$F2F-,&*&\"\"%F 1F*F1F1*&\"\"(F1F2F1F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 20 "com ma triz associada" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "A2 := genmatrix( \+ map(rhs,sys2), [x(t),y(t)], `-b` );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#A2GK%'matrixG6#7$7$\"\"$!\"\"7$\"\"%\"\"(Q(pprint06\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 94 "e autovalores e autovetores como abaixo \+ - note que n\343o existem dois autovetores independentes!" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 22 "z2:= eigenvectors(A2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z2G7%\"\"&\"\"#<#K%'vectorG6#7$\"\"\"!\"#Q(pprint06 \"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "Vamos pedir a solu\347\343o geral, para ver que ela n\343o \351 uma simples combina\347\343o de e xponenciais puras.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "gen_soln := dsolve( sys2, \{x(t),y(t)\} );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%) gen_solnG<$/-%\"xG6#%\"tG*&-%$expG6#,$*&\"\"&\"\"\"F*F2F2F2,&%$_C1GF2* &%$_C2GF2F*F2F2F2/-%\"yGF),$*&F,F2,(*&\"\"#F2F4F2F2*(F>F2F6F2F*F2F2F6F 2F2!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 21 "Temos um sistema com " }{TEXT 230 73 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Como a parte real dos autovalores \351 posit iva, as espirais est\343o indo para infinito." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 339 "dir_field4 := DEplot( sys4, \{x(t),y(t)\}, t=0..1, x =-1..1, y=-3..3, scene=[x,y] ):\nIC4:=[ [x(0)=0.1*i,y(0)= 0.1*i] $ i=0 ..5 , [x(0)=-0.1*i,y(0)= -0.1*i] $ i=0..5]:\nphase_port4 := DEplot( sy s4, \{x(t),y(t)\}, t=-3..3,x=-1..1, y=-3..3, IC4, scene=[x,y],arrows=N ONE, linecolor=CYAN,stepsize=0.03 ):\ndisplay([dir_field4,phase_port4] , axes=BOXED);" }}{PARA 13 "" 1 "" {GLPLOT2D 593 593 593 {PLOTDATA 2 " 62-%'CURVESG6^dlX4%)anythingG6\"6\"[gl'!%\"!!#'\"$\"#!!!!!!!!BFF013F6F 0218A33C006DE836619AC17BFEFD8121FBCEB99C009217C99E653E9BFEF90D9D593590 7C00862C050453143X4F'F(F)[gl'!%\"!!#'\"$\"#!!!!!!!!BFF00E51B305F2A9C00 4570676867BCFBFEFE35C99F41AADC0069B8046057FEDBFEF94BE805C3D96C005DE683 8302A0FX4F'F(F)[gl'!%\"!!#'\"$\"#!!!!!!!!BFF00817DE11B6E5C001CFBF7FC8C 7FBBFEFEFD043DC9235C004154DF993E7ABBFEF992238FE0F2AC0035A340AA117B5X4F 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