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"Times" 1 18 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 48 "MAT1154 - Equa\347\365e s Diferenciais e de Diferen\347as " }}{PARA 256 "" 0 "" {TEXT 257 33 " Alguns Exerc\355cios da Lista No. 5." }}{PARA 256 "" 0 "" {TEXT -1 3 " por" }}{PARA 256 "" 0 "" {TEXT -1 17 "George Svetlichny" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "Inicializamo s com os pacotes de equa\347\365es diferenciais, plotagem, \351 transf ormadas integrais." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "resta rt:with(DEtools):with(plots):with(inttrans):" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 190 "Introduzimos um procedimento que \"limpa\" todas as de fini\347\365es anteriores. Vai ser usado no in\355cio de cada exerc \355cio para que as defini\347\365es dadas no exerc\355cio anterior n \343o tenham mais efeito.." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Limpar:=proc() global teq,teqi,lapsol,sol:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "unassign('f(t)'): unassign('g(t)'): unassign('c(t)') : unassign('y(t)'): unassign('Y(s)'): unassign('teq'): unassign('teqi' ): unassign('lapsol'): unassign('sol'):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "end proc:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "A pr\363xima \+ fun\347\343o ser\341 usada para plotar uma linha vertical para represe ntar a fun\347\343o impulso." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "vl:=(a,b)->[[a,0],[a,b]]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 330 11 "Exerc \355cio 1" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 " Calcu lo as transfomradas de Laplace" }}{PARA 0 "" 0 "" {TEXT -1 7 "(a) \+ " }{XPPEDIT 18 0 "f(t) = PIECEWISE([0, t < 2],[(t-2)^2, 2 <= t]);" "6# /-%\"fG6#%\"tG-%*PIECEWISEG6$7$\"\"!2F'\"\"#7$*$,&F'\"\"\"F.!\"\"F.1F. F'" }}{PARA 0 "" 0 "" {TEXT -1 7 "(b) " }{XPPEDIT 18 0 "f(t) = PIEC EWISE([0, t < Pi],[t-Pi, t <= 2*Pi],[0, 2*Pi < t]);" "6#/-%\"fG6#%\"tG -%*PIECEWISEG6%7$\"\"!2F'%#PiG7$,&F'\"\"\"F.!\"\"1F'*&\"\"#F1F.F17$F,2 *&F5F1F.F1F'" }}{PARA 0 "" 0 "" {TEXT -1 6 "(c) " }{XPPEDIT 18 0 "f( t) = t-Heaviside(t-1)*(t-1);" "6#/-%\"fG6#%\"tG,&F'\"\"\"*&-%*Heavisid eG6#,&F'F)F)!\"\"F),&F'F)F)F/F)F/" }{TEXT -1 6 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Not que em Maple, a \+ fun\347\343o salto " }{TEXT 259 17 "u_c(t) = u_0(t-c)" }{TEXT -1 21 " \+ \351 representado por " }{TEXT 258 14 "Heaviside(t-c)" }{TEXT -1 4 ". " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 260 17 "Resolu\347 \343o de (a):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Primeiro definim os a fun\347\343o como uma definida por partes:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 45 "Limpar():f(t):=piecewise(t<2,0,t>=2,(t-2)^2); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Plotamos a fun\347\343o:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(f(t),t=0..10);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 128 "O Maple n\343o manipula diretamen te fun\347\365es definidas por partes, \351 preciso convert\352-los em express\365es utilizando fun\347\365es de salto:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 30 "f(t):=convert(f(t),Heaviside);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "O comando " }{TEXT 261 7 "laplace" } {TEXT -1 35 " calcula a transformada de laplace." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "laplace(f(t),t,s);" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }{TEXT 262 17 "Resolu\347\343o de (b):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Definimos a fun\347\343o como uma definid a por partes:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " " }{TEXT -1 0 "" }{MPLTEXT 1 0 55 "Limpar():f(t):=piecewise(t2*Pi,0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Plotamos a fun\347 \343o:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(f(t),t=0..10 );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Convertemos para uma expres s\343o utilizando fun\347\365es salto." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f(t):=convert(f(t),Heaviside);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Calculamos a transformada de Laplace." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "laplace(f(t),t,s);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 12 "Resolu\347\343o de" }{TEXT -1 1 " " } {TEXT 263 4 "(c):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "A fun\347\343 o " }{TEXT 265 4 "f(t)" }{TEXT -1 32 " j\341 \351 dada usando fun\347 \365es salto." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 " Limpar(): " }{TEXT -1 0 "" }{MPLTEXT 1 0 29 "f(t):=t-Heaviside(t-1)*(t-1);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Plotando a fun\347\343o." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(f(t),t=0..10);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Calculando a transformada de Lapla ce." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "laplace(f(t),t,s);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 331 12 "Exerc\355cio 2 " }{TEXT -1 0 "" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Resolva os problemas de valor inci al:" }}{PARA 0 "" 0 "" {TEXT -1 8 "(a) " }{XPPEDIT 18 0 "`@@`(D,2) (y)(t)+y(t) = f(t);" "6#/,&---%#@@G6$%\"DG\"\"#6#%\"yG6#%\"tG\"\"\"-F- 6#F/F0-%\"fG6#F/" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = 0;" "6#/-% \"yG6#\"\"!F'" }{TEXT -1 4 " e " }{XPPEDIT 18 0 "D(y)(0) = 1;" "6#/-- %\"DG6#%\"yG6#\"\"!\"\"\"" }{TEXT -1 6 ", e " }{XPPEDIT 18 0 "f(t) = PIECEWISE([1, t < 1/2*Pi],[0, 1/2*Pi <= t]);" "6#/-%\"fG6#%\"tG-%*PIE CEWISEG6$7$\"\"\"2F'*(F,F,\"\"#!\"\"%#PiGF,7$\"\"!1*(F,F,F/F0F1F,F'" } }{PARA 0 "" 0 "" {TEXT -1 8 "(b) " }{XPPEDIT 18 0 "`@@`(D,2)(y)(t) +4*y(t) = sin(t)-Heaviside(t-2)*sin(t-2*Pi);" "6#/,&---%#@@G6$%\"DG\" \"#6#%\"yG6#%\"tG\"\"\"*&\"\"%F0-F-6#F/F0F0,&-%$sinG6#F/F0*&-%*Heavisi deG6#,&F/F0F+!\"\"F0-F76#,&F/F0*&F+F0%#PiGF0F>F0F>" }{TEXT -1 4 ", \+ " }{XPPEDIT 18 0 "y(0) = 0;" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 4 " e " } {XPPEDIT 18 0 "D(y)(0) = 0;" "6#/--%\"DG6#%\"yG6#\"\"!F*" }}{PARA 0 " " 0 "" {TEXT -1 8 "(c) " }{XPPEDIT 18 0 "`@@`(D,2)(y)(t)+2*D(y)(t) +2*y(t) = Dirac(t-Pi);" "6#/,(---%#@@G6$%\"DG\"\"#6#%\"yG6#%\"tG\"\"\" *&F+F0--F*6#F-6#F/F0F0*&F+F0-F-6#F/F0F0-%&DiracG6#,&F/F0%#PiG!\"\"" } {TEXT -1 4 ", " }{XPPEDIT 18 0 "y(0) = 1;" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 4 " e " }{XPPEDIT 18 0 "D(y)(0) = 0;" "6#/--%\"DG6#%\"yG6# \"\"!F*" }}{PARA 0 "" 0 "" {TEXT -1 7 "(d) " }{XPPEDIT 18 0 "`@@`(D ,2)(y)(t)+w*y(t) = Dirac(t-Pi/w);" "6#/,&---%#@@G6$%\"DG\"\"#6#%\"yG6# %\"tG\"\"\"*&%\"wGF0-F-6#F/F0F0-%&DiracG6#,&F/F0*&%#PiGF0F2!\"\"F;" } {TEXT -1 5 ", " }{XPPEDIT 18 0 "y(0) = 0;" "6#/-%\"yG6#\"\"!F'" } {TEXT -1 4 " e " }{XPPEDIT 18 0 "D(y)(0) = 1;" "6#/--%\"DG6#%\"yG6#\" \"!\"\"\"" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "(e) " } {XPPEDIT 18 0 "`@@`(D,2)(y)(t)+4*y(t) = cos(t)+Dirac(t-Pi/2);" "6#/,&- --%#@@G6$%\"DG\"\"#6#%\"yG6#%\"tG\"\"\"*&\"\"%F0-F-6#F/F0F0,&-%$cosG6# F/F0-%&DiracG6#,&F/F0*&%#PiGF0F+!\"\"F?F0" }{TEXT -1 5 " , " } {XPPEDIT 18 0 "y(0) = 0;" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 4 " e " } {XPPEDIT 18 0 "D(y)(0) = 0;" "6#/--%\"DG6#%\"yG6#\"\"!F*" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 266 14 "Resolu\347 \343o (a):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Definimos a equa \347\343o, a fun\347\343o " }{TEXT 273 4 "f(t)" }{TEXT -1 46 " e conve rtemos a \372ltima usando fun\347\365es saltos." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 72 "Limpar():eq:=(D@@2)(y)(t)+y(t)=f(t);f(t):=piec ewise(t=Pi/2,0);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f(t): =convert(f(t),Heaviside);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Plota mos " }{TEXT 274 4 "f(t)" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(f(t),t=0..10);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Calculamos a transformada de Laplace dos dois lados da equa\347 \343o indicando por " }{TEXT 275 4 "Y(s)" }{TEXT -1 31 " a transforma da de laplace de " }{TEXT 276 4 "y(t)" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "teq:=subs(laplace(y(t),t,s)=Y(s),la place(eq,t,s));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Introduzimos o s valores inciais na express\343o acima." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "teqi:=subs(y(0)=0,D(y)(0)=1,teq);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Resolvemos para " }{TEXT 277 4 "Y(s)" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Y(s):=solve(teqi, Y (s));" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Calculamo s a transformada de laplace inversa (commando " }{TEXT 278 10 "invlapl ace" }{TEXT -1 5 ") de " }{TEXT 279 4 "Y(s)" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "y(t):=invlaplace(Y(s),s,t); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Plotamo a solu\347\343o (em v ermelho) junto com o termo n\343o homog\352neo (em azul)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "plot([f(t),y(t)],t=0..10,color=[blu e,red]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "A solu\347\343o parec e uma curva senoidal " }{XPPEDIT 307 0 "2*sin(t)-cos(t);" "6#,&*&\"\"# \"\"\"-%$sinG6#%\"tGF&F&-%$cosG6#F*!\"\"" }{TEXT -1 56 ", mas n\343o \+ \351. Vamos express\341-la como definida por partes:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "convert(y(t),piecewise);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "Para vizualizar isto melhor, plotamos ag ora o termo n\343o homog\352neo (azul), a solu\347\343o (vermelho), e \+ a fun\347\343o " }{XPPEDIT 308 0 "2*sin(t)-cos(t);" "6#,&*&\"\"#\"\" \"-%$sinG6#%\"tGF&F&-%$cosG6#F*!\"\"" }{TEXT -1 11 " (verde). " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "plot([f(t),y(t),2*sin(t)-cos (t)],t=0..10,color=[blue,red,green]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 267 14 "Resolu\347\343o (b):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Defin imos a equa\347\343o e a fun\347\343o " }{TEXT 280 4 "f(t)" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "Limpar(): eq:=(D@@2 )(y)(t)+4*y(t)=f(t); f(t):=sin(t)-Heaviside(t-2)*sin(t-2*Pi);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Plotamos " }{TEXT 285 4 "f(t)" } {TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(f(t), t=0..10);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Calculamos a transfo rmada de Laplace dos dois lados da equa\347\343o indicando por " } {TEXT 287 4 "Y(s)" }{TEXT -1 31 " a transformada de laplace de " } {TEXT 288 4 "y(t)" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "teq:=subs(laplace(y(t),t,s)=Y(s),laplace(eq,t,s));" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Introduzimos os valores inciais \+ na express\343o acima." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "t eqi:=subs(y(0)=0,D(y)(0)=0,teq);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Resolvemos para " }{TEXT 295 4 "Y(s)" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Y(s):=solve(teqi, Y(s));" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Calculamos a transforma da de laplace inversa (commando " }{TEXT 299 10 "invlaplace" }{TEXT -1 5 ") de " }{TEXT 300 4 "Y(s)" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 27 "y(t):=invlaplace(Y(s),s,t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Plotamo a solu\347\343o (em vermelho) junto com o termo n\343o homog\352neo (em azul)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "plot([f(t),y(t)],t=0..10,color=[blue,red]);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Expressamos a solu\347\343o como \+ uma fun\347\343o definida por partes:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "convert(y(t),piecewise);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 268 14 "Resolu\347\343o (c):" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 31 "Definimos a equa\347\343o e a fun\347\343o " } {TEXT 281 4 "f(t)" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "Limpar(): eq:=(D@@2)(y)(t)+2*D(y)(t)+2*y(t)=f(t); f(t ):=Dirac(t-Pi);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Plotamos " } {TEXT 286 4 "f(t)" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot(vl(Pi,5),t=0..5);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Calculamos a transformada de Laplace dos dois lados da eq ua\347\343o indicando por " }{TEXT 289 4 "Y(s)" }{TEXT -1 31 " a tran sformada de laplace de " }{TEXT 290 4 "y(t)" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "teq:=subs(laplace(y(t),t,s)= Y(s),laplace(eq,t,s));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Introdu zimos os valores inciais na express\343o acima." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 33 "teqi:=subs(y(0)=1,D(y)(0)=0,teq);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Resolvemos para " }{TEXT 296 4 "Y(s)" } {TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Y(s):=solv e(teqi, Y(s));" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 " Calculamos a transformada de laplace inversa (commando " }{TEXT 301 10 "invlaplace" }{TEXT -1 5 ") de " }{TEXT 302 4 "Y(s)" }{TEXT -1 1 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "y(t):=invlaplace(Y(s),s ,t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Plotamo a solu\347\343o ( em vermelho) junto com o termo n\343o homog\352neo (em azul)." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "plot([vl(Pi,1),y(t)],t=0..15 ,y=-0.5..1,color=[blue,red]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 269 14 "Resolu\347\343o (d):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Definimo s a equa\347\343o e a fun\347\343o " }{TEXT 282 5 "f(t)." }{TEXT -1 27 " Declaramos que a vari\341vel " }{TEXT 283 1 "w" }{TEXT -1 31 " re presenta um n\372mero positivo." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "Limpar():assume(w>0):eq:=(D@@2)(y)(t)+w*y(t)=f(t); f(t):=Dirac (t-Pi/w);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "O til que segue a va ri\341vel " }{TEXT 310 1 "w" }{TEXT -1 92 " indica que o que ela repr esenta tem uma propriedade assumida (ser positivo no nosso caso)." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Calculamos a transformada de Lapla ce dos dois lados da equa\347\343o indicando por " }{TEXT 291 4 "Y(s) " }{TEXT -1 31 " a transformada de laplace de " }{TEXT 292 4 "y(t)" } {TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "teq:=subs( laplace(y(t),t,s)=Y(s),laplace(eq,t,s));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Introduzimos os valores inciais na express\343o acima. (A vari\341vel " }{TEXT 309 1 "w" }{TEXT -1 22 " foi substituido por " }{XPPEDIT 18 0 "omega;" "6#%&omegaG" }{TEXT -1 47 " para contornar um \+ certo problema com o Maple)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "teqi:=subs(y(0)=0,D(y)(0)=1,w=omega,teq);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Resolvemos para " }{TEXT 297 4 "Y(s)" }{TEXT -1 1 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Y(s):=solve(teqi, Y(s)) ;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Calculamos a \+ transformada de laplace inversa (commando " }{TEXT 303 10 "invlaplace " }{TEXT -1 5 ") de " }{TEXT 304 4 "Y(s)" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "y(t):=invlaplace(Y(s),s,t);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Escolhendo " }{XPPEDIT 18 0 "omeg a = Pi;" "6#/%&omegaG%#PiG" }{TEXT -1 78 ", plotamos a solu\347\343o \+ (em vermelho) junto com o termo n\343o homog\352neo (em azul)." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "omega:=Pi:plot([vl(Pi/omega, 1),y(t)],t=0..10,y=-1..1,color=[blue,red]);" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 38 "Um fen\364meno interessante acontece com " }{XPPEDIT 18 0 "omega = 1.;" "6#/%&omegaG$\"\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "omega:=1:plot([vl(Pi/omega,1),y(t)],t=0..10,y=-1.. 1,color=[blue,red]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "A \"panca da\" em " }{XPPEDIT 18 0 "t = Pi;" "6#/%\"tG%#PiG" }{TEXT -1 39 " p \341ra totalmente o movimento inicial. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 270 14 "Resolu\347 \343o (e):" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Defi nimos a equa\347\343o e a fun\347\343o " }{TEXT 284 4 "f(t)" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "Limpar(): eq:=(D @@2)(y)(t)+2*D(y)(t)+2*y(t)=f(t); f(t):=cos(t)+Dirac(t-Pi/2);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "plot([vl(Pi/2,1),f(t)],t=0.. 10,color=[red,red]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Calculamo s a transformada de Laplace dos dois lados da equa\347\343o indicando \+ por " }{TEXT 293 4 "Y(s)" }{TEXT -1 31 " a transformada de laplace de " }{TEXT 294 4 "y(t)" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "teq:=subs(laplace(y(t),t,s)=Y(s),laplace(eq,t,s));" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Introduzimos os valores inciais \+ na express\343o acima." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "t eqi:=subs(y(0)=0,D(y)(0)=0,teq);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Resolvemos para " }{TEXT 298 4 "Y(s)" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Y(s):=solve(teqi, Y(s));" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Calculamos a transforma da de laplace inversa (commando " }{TEXT 305 10 "invlaplace" }{TEXT -1 5 ") de " }{TEXT 306 4 "Y(s)" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 27 "y(t):=invlaplace(Y(s),s,t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Plotamo a solu\347\343o (em vermelho) junto com o termo n\343o homog\352neo (em azul)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "plot([f(t),vl(Pi/2,1),y(t)],t=0..20,color=[blue,blue, red]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Expressamos a solu\347 \343o como uma fun\347\343o definida por partes:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 24 "convert(y(t),piecewise);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 332 11 " Exerc\355cio 3" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "U sando Transformada de Laplace, determine a solu\347\343o do seguinte p roblema de valor inicial:" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "D(y)(t)- 2*y(t) = 2*exp(-2*t);" "6#/,&--%\"DG6#%\"yG6#%\"tG\"\"\"*&\"\"#F,-F)6# F+F,!\"\"*&F.F,-%$expG6#,$*&F.F,F+F,F1F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "0 <= t;" "6#1\"\"!%\"tG" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "y(1) = 0;" "6#/-%\"yG6#\"\"\"\"\"!" }{TEXT -1 2 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Definimos a equa\347\343o." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Limpar(): eq:=D(y)(t)-2*y(t)=2*exp(-2*t);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Calculamos a transformada de Lapla ce dos dois lados da equa\347\343o indicando por " }{TEXT 311 4 "Y(s) " }{TEXT -1 31 " a transformada de laplace de " }{TEXT 312 4 "y(t)" } {TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "teq:=subs( laplace(y(t),t,s)=Y(s),laplace(eq,t,s));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Como a condi\347\343o incial \351 no ponto " }{TEXT 313 3 "t=1" }{TEXT -1 3 " e " }{TEXT 314 4 "n\343o " }{TEXT -1 3 "em " } {TEXT 315 3 "t=0" }{TEXT -1 260 ", n\343o podemos substituir o valor i nicial nesta equa\347\343o. Vamos portanto usar a transformada de Lapl ace apenas para achar uma solu\347\343o particular da equa\347\343o pa ra depois somar con a solu\347\343o geral da equa\347\343o homog\352ne a e s\363 ent\343o impor a condi\347\343o incial. Escolhemos " }{TEXT 316 7 "y(0)=0 " }{TEXT -1 70 "para a nossa solu\347\343o particular (q ualquer outro valor pode ser usado)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "teqi:=subs(y(0)=0,teq);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Resolvemos para " }{TEXT 317 4 "Y(s)" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Y(s):=solve(teqi, Y(s));" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Calculamos a tran sformada de laplace inversa (commando " }{TEXT 318 10 "invlaplace" } {TEXT -1 5 ") de " }{TEXT 319 5 "Y(s) " }{TEXT -1 25 "o que nos d\341 \+ uma solu\347\343o " }{TEXT 321 10 "particular" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "solp:=invlaplace(Y(s),s,t); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Plotamos esta solu\347\343o p articular." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(solp,t=- 1..3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Definimos agora a equa \347\343o homog\352nea." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 " Limpar():eqh:=D(y)(t)-2*y(t)=0;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "Achamos a solu\347\343o geral da equa\347\343o homog\352nea, n\343 o usando a transformada de Laplace." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "solh:=subs(dsolve(eqh,y(t)),y(t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 124 "Somamos a solu\347\343o particular com a solu \347\343o geral da equa\347\343o homog\352nea. Isto nos d\341 a solu \347\343o geral da equa\347\343o n\343o homog\352nea." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "solgnh:=solp+solh;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Impomos a condi\347\343o inicial em " } {TEXT 320 3 "t=1" }{TEXT -1 42 " e resolvemos para a constante arbitr \341ria." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "C1:=simplify(ex pand(solve(subs(t=1,solgnh)=0,_C1)));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "Substituimos esta constante na solu\347\343o geral o que \+ nos d\341 a solu\347\343o procurada." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "y(t):=simplify(subs(_C1=C1,solgnh));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Plotamos a solu\347\343o." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(y(t),t=-1..3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 0 "" }{TEXT 333 11 "Exerc\355cio 4" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Consider o problema de valor inicial" }} {PARA 256 "" 0 "" {XPPEDIT 18 0 "`@@`(D,2)(y)(t)-4*y = g(t);" "6#/,&-- -%#@@G6$%\"DG\"\"#6#%\"yG6#%\"tG\"\"\"*&\"\"%F0F-F0!\"\"-%\"gG6#F/" } {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "y(0) = 0,D(y)(0) = 0; " "6$/-%\"yG6#\"\"!F'/--%\"DG6#F%6#F'F'" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "onde a fun\347\343o " }{TEXT 271 4 "g(t)" }{TEXT -1 12 " \351 da forma:" }}{PARA 0 "" 0 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7]p7$$\"\"!F)%%FAILG7$$\"3ALL$3FWYs#!#?$\" \"\"F)7$$\"3WmmmT&)G\\aF.F/7$$\"3m****\\7G$R<)F.F/7$$\"3ILLL3x&)*3\"!# >F/7$$\"3$*****\\ilyM;F:F/7$$\"3emmm;arz@F:F/7$$\"3')*****\\7t&pKF:F/7 $$\"39LLLL3VfVF:F/7$$\"3s******\\i9RlF:F/7$$\"3Hmmmm;')=()F:F/7$$\"3-+ +]7z>^7!#=F/7$$\"3RLLLe'40j\"FPF/7$$\"3mmmm;6m$[#FPF/7$$\"3fmmm;yYULFP F/7$$\"3%HLL$eF>(>%FPF/7$$\"3Qmmm\">K'*)\\FPF/7$$\"3P*****\\Kd,\"eFPF/ 7$$\"3-mmm\"fX(emFPF/7$$\"3.*****\\U7Y](FPF/7$$\"3'QLLLV!pu$)FPF/7$$\" 3K+++DI(yv)FPF/7$$\"3xmmm;c0T\"*FPF/7$$\"3))****\\P?uc$*FPF/7$$\"3+LLL e%GCd*FPF/7$$\"37++vo;F!o*FPF/7$$\"37mm;z[6)y*FPF/7$$\"3d)*\\P%[O?%)*F PF/7$$\"37KLe*3ef*)*FPF/7$$\"3!*)\\(=#*)=H#**FPF/7$$\"3ol;z%pz)\\**FPF /7$$\"3YKeR(\\So(**FPF/7$$\"3#*******H,Q+5!#<$!\"\"F)7$$\"3&*******RXp V5FjqF[r7$$\"3)*******\\*3q3\"FjqF[r7$$\"3)*******p=\\q6FjqF[r7$$\"3mm m;fBIY7FjqF[r7$$\"3GLLLj$[kL\"FjqF[r7$$\"3?LLL`Q\"GT\"FjqF[r7$$\"3!*** **\\s]k,:FjqF[r7$$\"39LLL`dF!e\"FjqF[r7$$\"33++]sgam;FjqF[r7$$\"3/++]< ep[FjqF[r7$$\"3)****\\([W db>FjqF[r7$$\"3immmTc-)*>FjqF[r7$$\"3lT&Q.d\"y+?FjqF(7$$\"3n;/,*\\PN+# FjqF(7$$\"3p\"H#oFMH1?FjqF(7$$\"3qmTNc$\\!4?FjqF(7$$\"3u;zp87c9?FjqF(7 $$\"3ym;/rI2??FjqF(7$$\"3&o;Hdy'4J?FjqF(7$$\"3[mmT+07U?FjqF(7$$\"3=m;z Hz;k?FjqF(7$$\"3Mmm;f`@'3#FjqF(7$$\"3HLLLj+gC@FjqF(7$$\"3y****\\nZ)H;# FjqF(7$$\"3YmmmJy*eC#FjqF(7$$\"3')******R^bJBFjqF(7$$\"3f*****\\5a`T#F jqF(7$$\"3o****\\7RV'\\#FjqF(7$$\"3k*****\\@fke#FjqF(7$$\"3/LLL`4NnEFj qF(7$$\"3#*******\\,s`FFjqF(7$$\"3[mm;zM)>$GFjqF(7$$\"3$*******pfa " 0 "" {MPLTEXT 1 0 90 "Limpar():eq:=(D@@2 )(y)(t)-4*y(t)=g(t); g(t):=Heaviside(t)-2*Heaviside(t-1)+Heaviside(t-2 );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Calculamos a transformada d e Laplace de " }{TEXT 323 5 "g(t)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "laplace(g(t),t,s);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 353 13 "Resolu\347\343o (b)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Calculamos a transformada de Laplace dos dois lados da eq ua\347\343o indicando por " }{TEXT 324 4 "Y(s)" }{TEXT -1 31 " a tran sformada de laplace de " }{TEXT 325 4 "y(t)" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "teq:=subs(laplace(y(t),t,s)= Y(s),laplace(eq,t,s));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Introdu zimos os valores inciais na express\343o acima." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 33 "teqi:=subs(y(0)=0,D(y)(0)=0,teq);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Resolvemos para " }{TEXT 326 4 "Y(s)" } {TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Y(s):=solv e(teqi, Y(s));" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 " Calculamos a transformada de laplace inversa (commando " }{TEXT 327 10 "invlaplace" }{TEXT -1 5 ") de " }{TEXT 328 4 "Y(s)" }{TEXT -1 1 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "y(t):=invlaplace(Y(s),s ,t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Plotamo a solu\347\343o ( em vermelho) junto com o termo n\343o homog\352neo (em azul)." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "plot([g(t),y(t)],t=0..2.5,co lor=[blue,red]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "Expressamos a solu\347\343o como uma fun\347\343o definida por partes. Limpamos ta mbem a defini\347\343o de " }{TEXT 329 4 "g(t)" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "convert(y(t),piecewise);unas sign('g(t)'):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 11 "Exerc\355cio 7" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "(a) Calcule " }{TEXT 334 4 "w(t)" }{TEXT -1 37 " a transformada de Laplace inversa de" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "W(s) = exp(-2*s)/(s*(s^2-1));" "6#/-%\"WG6#%\"sG* &-%$expG6#,$*&\"\"#\"\"\"F'F/!\"\"F/*&F'F/,&*$F'F.F/F/F0F/F0" }{TEXT -1 3 " \n" }}{PARA 0 "" 0 "" {TEXT -1 65 "(b) Ache a solu\347\343o da equa\347\343o y''-y=u_2(t) que satisfaz y(0)=0 e " }{XPPEDIT 18 0 "li mit(y(t),t = infinity) = 1;" "6#/-%&limitG6$-%\"yG6#%\"tG/F*%)infinity G\"\"\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 335 13 "Resolu\347\343o (a)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "O comando " }{TEXT 336 10 "invlaplace" }{TEXT -1 42 " calcula a tr ansformada de Laplace inversa" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "w(t):=invlaplace(exp(-2*s)/(s*(s^2-1)),s,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "plot(w(t),t=0..8);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 337 13 "Resolu\347\343o (b)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Definimos a equa\347\343o em quest\343o:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Limpar():eq:=(D@@2)(y)(t)-y( t)=Heaviside(t-2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Neste ponto devemos reconhecer que a fun\347\343o " }{TEXT 341 4 "W(s)" }{TEXT -1 67 " \351 a transformada de Laplace da solu\347\343o desta equa\347 \343o que satisfaz " }{TEXT 342 6 "y(0)=0" }{TEXT -1 3 " e " }{TEXT 343 7 "y'(0)=0" }{TEXT -1 23 ". Vamos verificar isto." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Calculamos a transformada de Laplace dos \+ dois lados da equa\347\343o indicando por " }{TEXT 338 4 "Y(s)" } {TEXT -1 31 " a transformada de laplace de " }{TEXT 339 4 "y(t)" } {TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "teq:=subs( laplace(y(t),t,s)=Y(s),laplace(eq,t,s));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Introduzimos os valores inciais na express\343o acima." } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "teqi:=subs(y(0)=0,D(y)(0)= 0,teq);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Resolvemos para " } {TEXT 340 4 "Y(s)" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Y(s):=solve(teqi, Y(s));" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Reconhecemos aqui a fun\347\343o " } {TEXT 354 4 "W(s)" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Consideramos agora a equa\347\343o homog\352nea." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "eqh:=(D@@2)(y)(t)-y(t)=0;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Achamos a solu\347\343o geral da equa\347 \343o homog\352nea:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "solh :=dsolve(eqh,y(t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "A solu\347 \343o geral da n\343o homog\352nea \351 a soma da solu\347\343o partic ular " }{TEXT 344 4 "w(t)" }{TEXT -1 34 " com a solu\347\343o geral da homog\352nea." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "solgnh:=s ubs(solh,y(t))+w(t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Precisamo s agora impor as condi\347\365es em " }{TEXT 345 2 "0 " }{TEXT -1 2 "e " }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT -1 24 ". Calculam os o valor em " }{TEXT 346 2 "0." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "simplify(subs(t=0,solgnh));" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 13 "Assim temos " }{XPPEDIT 18 0 "_C1+_C2 = 0;" "6#/,&%$_C 1G\"\"\"%$_C2GF&\"\"!" }{TEXT -1 47 ". Para ver a condi\347\343o no in finito reescrevemos " }{TEXT 347 4 "w(t)" }{TEXT -1 22 " em forma expo nencial." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "simplify(conver t(w(t),exp));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Para " }{TEXT 348 5 "t -> " }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT -1 2 " , " }{TEXT 349 19 "Heaviside(t-2) = 1 " }{TEXT -1 19 "e portanto o ter mo " }{XPPEDIT 18 0 "_C1;" "6#%$_C1G" }{TEXT -1 22 " deve cancelar o t ermo" }{TEXT 350 1 " " }{XPPEDIT 351 0 "1/2*Heaviside(t-2)*exp(t-2);" "6#**\"\"\"F$\"\"#!\"\"-%*HeavisideG6#,&%\"tGF$F%F&F$-%$expG6#,&F+F$F% F&F$" }{TEXT -1 11 ". Assim " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "_C1 \+ = -exp(-2)/2;" "6#/%$_C1G,$*&-%$expG6#,$\"\"#!\"\"\"\"\"F+F,F," } {TEXT -1 7 " e " }{XPPEDIT 18 0 "_C2 = exp(-2)/2;" "6#/%$_C2G*&-%$ expG6#,$\"\"#!\"\"\"\"\"F*F+" }{TEXT -1 21 " . A nossa solu\347\343o \+ \351:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "y(t):=subs(_C1 = - exp(-2)/2,_C2 = exp(-2)/2,solgnh);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Vamos verificar se as condi\347\365es s\343o satisfeitas." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "simplify(subs(t=0,y(t)));" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "limit(y(t),t=infinity);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "Plotamos a solu\347\343o encontrad a (vermelho) e o termo n\343o homog\352neo (azul)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "plot([Heaviside(t-2),y(t)],t=-3..20,color=[ blue,red]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "12" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }