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"MAT1154 \+ - Equa\347\365es Diferenciais e de Diferen\347as " }}{PARA 257 "" 0 " " {TEXT -1 65 "Equa\347\365es diferenciais e de diferen\347as lineare s de segunda ordem." }{TEXT 327 0 "" }}{PARA 256 "" 0 "" {TEXT -1 19 " A fun\347\343o de Dirac " }{XPPEDIT 258 0 "delta(t);" "6#-%&deltaG6#% \"tG" }{TEXT -1 17 " e a convolu\347\343o." }}{PARA 256 "" 0 "" {TEXT -1 3 "por" }}{PARA 256 "" 0 "" {TEXT -1 17 "George Svetlichny" } }}{EXCHG {PARA 259 "" 0 "" {TEXT 330 81 "O objetivo desta planilha \+ \351 apresentar algumas propriedades principais da fun\347\343o " } {TEXT 329 9 "de Dirac " }{XPPEDIT 268 0 "delta(t);" "6#-%&deltaG6#%\"t G" }{TEXT 333 48 " (fun\347\343o impulso) e da opera\347\343o de convo lu\347\343o. " }{TEXT 339 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "restart:with(DEtools):w ith(plots):with(inttrans):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Int roduzimos uma fun\347\343o " }{XPPEDIT 18 0 "delta[a](t)" "6#-&%&delta G6#%\"aG6#%\"tG" }{TEXT -1 10 " que vale " }{TEXT 256 4 "1/a " }{TEXT -1 13 "no intervalo " }{TEXT 257 5 "(0,a)" }{TEXT -1 34 " e zero fora. Isto seria a fun\347\343o " }{XPPEDIT 258 0 "(1-u[a](t))/a;" "6#*&,& \"\"\"F%-&%\"uG6#%\"aG6#%\"tG!\"\"F%F*F-" }{TEXT -1 34 " na nota\347 \343o de Boyce & DiPrima. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "assume(a>0):delta[a]:=(a,t)->(Heaviside(t)-Heaviside(t-a))/a;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Vamos calcular a integral da nossa fun\347\343o. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "int(delt a[a](a,t),t=0..infinity);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Assi m a \341rea em baixo do gr\341fico de " }{XPPEDIT 18 0 "delta[a](t);" "6#-&%&deltaG6#%\"aG6#%\"tG" }{TEXT -1 10 " \351 sempre " }{TEXT 260 1 "1" }{TEXT -1 30 " qualquer que seja o valor de " }{TEXT 261 1 "a" } {TEXT -1 27 ". Devemos agora pensar que " }{TEXT 259 1 "a" }{TEXT -1 89 " fica cada vez menor. Deste jeito a \341rea fica cada vez mais con centrada ao lado do ponto " }{TEXT 262 3 "t=0" }{TEXT -1 58 ". Vamos a gora plotar estas fun\347\365es para v\341rios valores de " }{TEXT 263 1 "a" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "plot([delta[a](1,t),delta[a](1/2,t),delta[a](1/3,t),delta[a](1/10 ,t)],t=0..5,color=[yellow,green,blue,red],legend=[\"a=1\",\"a=1/2\",\" a=1/3\",\"a=1/10\"]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Calculam os agora a transformada de Laplace de " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "Ldelta[a]:=simplify(laplace(delta[a](a,t),t,s));" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Plotamos esta transformada de Lapl ace para v\341rios valores de " }{TEXT 264 1 "a" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "plot([subs(a=1,Ldelta[a]),s ubs(a=1/10,Ldelta[a]),subs(a=1/100,Ldelta[a]),subs(a=1/1000,Ldelta[a]) ],s=0..100,color=[yellow,green,blue,red],legend=[\"a=1\",\"a=1/10\",\" a=1/100\",\"a=1/1000\"]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Apes ar da fun\347\343o " }{XPPEDIT 18 0 "delta[a](t)" "6#-&%&deltaG6#%\"aG 6#%\"tG" }{TEXT -1 27 " n\343o possuir limite quando " }{TEXT 265 4 "a ->0" }{TEXT -1 37 ", a transfomad de Laplace o possui: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "F:=limit(Ldelta[a],a=0,right);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "O limite simb\363lico de " } {XPPEDIT 18 0 "delta[a](t)" "6#-&%&deltaG6#%\"aG6#%\"tG" }{TEXT -1 25 " \351 conhecido como fun\347\343o " }{XPPEDIT 18 0 "delta;" "6#%&delt aG" }{TEXT -1 20 " de Dirac, ou ainda " }{TEXT 266 15 "fun\347\343o im pulso." }{TEXT -1 53 " A transformada de Laplace dela \351 a fun\347 \343o constante " }{TEXT 267 3 "1. " }{TEXT -1 42 "No Maple a fun\347 \343o impulso \351 designado por " }{TEXT 268 8 "Dirac(t)" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "invlaplace(F,s,t); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Vamos fazer a mesma coisa em \+ torno de " }{TEXT 269 3 "t=2" }{TEXT -1 9 ", isto \351 " }{XPPEDIT 18 0 "delta[a](t-2);" "6#-&%&deltaG6#%\"aG6#,&%\"tG\"\"\"\"\"#!\"\"" } {TEXT -1 4 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 156 "plot([ delta[a](1,t-2),delta[a](1/2,t-2),delta[a](1/3,t-2),delta[a](1/10,t-2) ],t=0..5,color=[yellow,green,blue,red],legend=[\"a=1\",\"a=1/2\",\"a=1 /3\",\"a=1/10\"]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "A transform ad de Laplace agora \351:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "Ldelta2[a]:=simplify(laplace(delta[a](a,t-2),t,s));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Como \351 de esperar, isto \351 a transfo rmada de Laplace de " }{XPPEDIT 18 0 "delta[a](t)" "6#-&%&deltaG6#%\"a G6#%\"tG" }{TEXT -1 18 " multiplicado por " }{XPPEDIT 18 0 "exp(-2*s); " "6#-%$expG6#,$*&\"\"#\"\"\"%\"sGF)!\"\"" }{TEXT -1 13 ". Assim com \+ " }{TEXT 270 4 "a->0" }{TEXT -1 19 ", isto converge ao " }{XPPEDIT 18 0 "exp(-2*s);" "6#-%$expG6#,$*&\"\"#\"\"\"%\"sGF)!\"\"" }{TEXT -1 69 " . Vamos primeiro plotar a fun\347\343o esta fun\347\343o para v\341ri os valores de " }{TEXT 293 1 "a" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 172 "plot([subs(a=2,Ldelta2[a]),subs(a=1,Ldelta2[a ]),subs(a=1/2,Ldelta2[a]),subs(a=1/10,Ldelta2[a])],s=0..2,color=[yello w,green,blue,red],legend=[\"a=2\",\"a=1\",\"a=1/2\",\"a=1/10\"]);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Verificamos o limite." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "F:=limit(Ldelta2[a],a=0,right);" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Isto \351 a transformada de Lapla ce de:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "invlaplace(F,s,t) ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Vamos agora plotar, para v \341rios valores de " }{TEXT 294 1 "a" }{TEXT -1 51 ", as solu\347\365 es do seguinte problema de valor inicial" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`@@`(D,2)(y)(t)+D(y)(t)+17/4*y(t) = delta[a] (t-2)" "6#/,(---%#@@G6$%\"DG\"\"#6#%\"yG6#%\"tG\"\"\"--F*6#F-6#F/F0*( \"# " 0 "" {MPLTEXT 1 0 54 "eq1:=(D@@2)(y)(t)+D(y)(t)+(17/4)*y(t)=delta[a](1,t-2):" } {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "eq2:=(D@@2)(y)(t)+D (y)(t)+(17/4)*y(t)=delta[a](1/2,t-2):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "eq3:=(D@@2)(y)(t)+D(y)(t)+(17/4)*y(t)=delta[a](1/3,t-2):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "eq4:=(D@@2)(y)(t)+D(y)(t)+(17/4)*y( t)=delta[a](1/10,t-2):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "sol1:=sub s(dsolve(\{eq1,y(0)=0,D(y)(0)=0\},y(t)),y(t)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "sol2:=subs(dsolve(\{eq2,y(0)=0,D(y)(0)=0\},y(t)),y(t) ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "sol3:=subs(dsolve(\{eq3,y(0)= 0,D(y)(0)=0\},y(t)),y(t)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "sol4: =subs(dsolve(\{eq4,y(0)=0,D(y)(0)=0\},y(t)),y(t)):" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 106 "plot([sol1,sol2,sol3,sol4],t=0..10,color=[yellow,g reen,blue,red],legend=[\"a=1\",\"a=1/2\",\"a=1/3\",\"a=1/10\"]);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "\311 f\341cil ver que as solu\347 \365es tendem a um limite. Este limite corresponde a tomar o termo n \343o homog\352neo igual a " }{XPPEDIT 272 0 "delta(t-2);" "6#-%&delta G6#,&%\"tG\"\"\"\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 61 "Vamos plotar esta solu\347\343o. Compare-o com os gr\341ficos a cima. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "eq0:=(D@@2)(y)(t )+D(y)(t)+(17/4)*y(t)=Dirac(t-2):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "sol0:=subs(dsolve(\{eq0,y(0)=0,D(y)(0)=0\},y(t)),y(t)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(sol0,t=0..10);" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 10 "A solu\347\343o " }{XPPEDIT 276 0 "K(t);" "6#-%\"K G6#%\"tG" }{TEXT -1 5 " de " }{XPPEDIT 277 0 "`@@`(D,2)(y)(t)+a*D(y)( t)+b*y(t) = delta(t)" "6#/,(---%#@@G6$%\"DG\"\"#6#%\"yG6#%\"tG\"\"\"*& %\"aGF0--F*6#F-6#F/F0F0*&%\"bGF0-F-6#F/F0F0-%&deltaG6#F/" }{TEXT -1 31 " que \351 identicamente zero para " }{TEXT 273 6 "t< 0, " }{TEXT -1 19 "\351 conhecido como a " }{TEXT 274 19 "solu\347\343o fundamenta l" }{TEXT 278 1 "." }{TEXT -1 68 " Ela \351 em primeiro lugar igual a \+ transformada de Laplace inversa de " }{XPPEDIT 18 0 "1/(s^2+a*s+b);" " 6#*&\"\"\"F$,(*$%\"sG\"\"#F$*&%\"aGF$F'F$F$%\"bGF$!\"\"" }{TEXT -1 9 " e para " }{TEXT 275 4 "t>0 " }{TEXT -1 27 "igual \340 solu\347\343o \+ da equa\347\343o " }{TEXT 279 9 "homog\352nea" }{TEXT -1 23 " com cond i\347\365es inicais " }{TEXT 280 6 "y(0)=0" }{TEXT -1 4 ", e " }{TEXT 281 9 "y'(0) = 1" }{TEXT 282 1 "." }{TEXT -1 94 " Vamos verificar este s fatos para a e.d.o. acima. Primeiro definimos a equa\347\343o n\343 o homog\352nea." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "eqnh:=(D @@2)(y)(t)+D(y)(t)+(17/4)*y(t)=Dirac(t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 111 "Resolvemos eata equa\347\343o usando a transformada de L aplace. Calculamos a transformada de Laplace dos dois lados.." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "teq:=subs(laplace(y(t),t,s)= Y(s),laplace(eqnh,t,s));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Subst ituimos os valores " }{TEXT 283 6 "y(0)=0" }{TEXT -1 4 ", e " }{TEXT 284 10 "y'(0) = 0." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "teqi: =subs(y(0)=0,D(y)(0)=0,teq);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "R esolvemos para " }{TEXT 285 5 "Y(s)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Y(s):=solve(teqi, Y(s));" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Calculamos a transformada de Laplace inve rsa." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "invlaplace(Y(s),s,t );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 104 "Na realidade, dado que, co m transformada de Laplace, n\363s trabalhamos somente com fun\347\365e s definidas para " }{XPPEDIT 18 0 "0 <= t;" "6#1\"\"!%\"tG" }{TEXT 286 2 " " }{TEXT -1 68 "a express\343o correta para K(t) seria aqu \352la acima multiplicada por " }{TEXT 296 12 "Heaviside(t)" }{TEXT -1 9 ", isto \351:" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }}{PARA 256 " " 0 "" {XPPEDIT 18 0 "Heaviside(t)*exp(-t/2)*sin(2*t)/2;" "6#**-%*Heav isideG6#%\"tG\"\"\"-%$expG6#,$*&F'F(\"\"#!\"\"F/F(-%$sinG6#*&F.F(F'F(F (F.F/" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 195 "Esta forma ser ia obtida do Maple se resolvermos a equa\347\343o diretamente sem pass ar pela transformada de Laplace. O \372nico cuidado que devemos tomar \+ \351 de impor as condi\347\365es inciais n\343o exatamente em " } {TEXT 287 3 "t=0" }{TEXT -1 55 " mas um pouco antes. Isto \351 porque \+ a fun\347\343o impulso em " }{TEXT 288 3 "t=0" }{TEXT -1 16 " faz o va lor de " }{TEXT 289 5 "y'(t)" }{TEXT -1 10 " mudar em " }{TEXT 290 3 " t=0" }{TEXT -1 58 " e, dependendo de como o software foi escrito, o va lor de " }{TEXT 291 6 "y'(0) " }{TEXT -1 80 "dado pelo usu\341rio pode ser interpretado de maneira imprevis'ivel pelo programa." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "solc:=dsolve(\{eqnh,y(-0.1)=0,D(y)( -0.1)=0\}, y(t)):K(t):=subs(solc,y(t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Isto \351 a resposta certa. Se colocarmos os dados inciai s em " }{TEXT 292 5 "t=0, " }{TEXT -1 9 "teriamos:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "sole:=dsolve(\{eqnh,y(0)=0,D(y)(0)=0\}, y(t ));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 113 "Isto difere da resposta c orreta por um termo que \351 a solu\347\343o da equa\347\343o homog \352nea. Vamos plotar as duas solu\347\365es." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "plot([K(t),subs(sole,y(t))],t=-2..10,color=[red ,blue],thickness=[2,2],legend=[\"K(t)\",\"Maple com y(0)=0,y'(0)=0\"]) ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Vamos agora resolver a equa\347\343o " }{TEXT 297 9 "homo g\352nea" }{TEXT -1 27 " com as condi\347oes iniciais " }{TEXT 298 6 " y(0)=0" }{TEXT -1 4 ", e " }{TEXT 299 11 "y'(0) = 1. " }{TEXT 300 0 " " }{TEXT -1 39 "Primeiro definimos a equa\347\343o homog\352nea." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "eqh:=(D@@2)(y)(t)+D(y)(t)+(1 7/4)*y(t)=0;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Resolvemos com as condi\347\365es iniciais." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "yh:=dsolve(\{eqh,y(0)=0,D(y)(0)=1\},y(t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Note que difere de " }{TEXT 301 4 "K(t)" }{TEXT -1 21 " por n\343o ter o fator " }{TEXT 302 12 "Heaviside(t)" }{TEXT -1 59 ". Vamos plotar juntos esta solu\347\343o e a solu\347\343o fundame ntal." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "plot([K(t),subs(y h,y(t))],t=-2..10,color=[red,blue],thickness=[2,3],legend=[\"K(t)\",\" h(t)=0,y(0)=0,y'(0)=1\"]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Not e que para " }{XPPEDIT 18 0 "0 <= t;" "6#1\"\"!%\"tG" }{TEXT -1 135 " \+ as duas solu\347\365es coincidem. Portanto podemos achar a solu\347 \343o fundamental simplesmente resolvendo a equa\347\343o homog\352nea com as condi\347\365es " }{TEXT 303 6 "y(0)=0" }{TEXT -1 4 ", e " } {TEXT 304 9 "y'(0) = 1" }{TEXT -1 17 " e lembrando que " }{TEXT 305 4 "K(t)" }{TEXT -1 27 " difere disto somente para " }{XPPEDIT 18 0 "t <= 0;" "6#1%\"tG\"\"!" }{TEXT -1 30 " onde \351 identicamente igual a " }{TEXT 306 1 "0" }{TEXT -1 3 ". " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "O seguinte porcedimento calcula a solu\347\343o fundamental de " } {XPPEDIT 18 0 "`@@`(D,2)(y)(t)+a*D(y)(t)+b*y(t) = delta(t);" "6#/,(--- %#@@G6$%\"DG\"\"#6#%\"yG6#%\"tG\"\"\"*&%\"aGF0--F*6#F-6#F/F0F0*&%\"bGF 0-F-6#F/F0F0-%&deltaG6#F/" }{TEXT -1 72 " resolvendo a equa\347\343o h omog\352nea correspondente. As respostas valem para " }{XPPEDIT 18 0 " 0 <= t;" "6#1\"\"!%\"tG" }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "solf:=proc(a,b) local eq,sol,K:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "eq:=(D@@2)(y)(t)+a*D(y)(t)+b*y(t)=0:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "sol:=dsolve(\{eq,y(0)=0,D(y)(0)=1\},y(t)):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "K:=subs(sol,y(t)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "end proc:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 124 "Vamos calcular e plotar algumas solu\347\365es fundamentais para \+ os exemplos j\341 apresentados na planinlha \"Segunda ordems linear.\" " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "K(t):=solf(-3,2);plot(K (t),t=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "K(t):=solf( 1,-2);plot(K(t),t=0..2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "K(t):=solf(1,1/4);plot(K(t),t=0..5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "K(t):=solf(1,17/4);plot(K(t),t=0..5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Sabendo a solu\347\343o fundamental " }{TEXT 307 0 "" }{TEXT -1 0 "" }{TEXT 308 4 "K(t)" }{TEXT -1 4 " de " }{XPPEDIT 256 0 "`@@`(D ,2)(y)(t)+a*D(y)(t)+b*y(t) = delta(t)" "6#/,(---%#@@G6$%\"DG\"\"#6#%\" yG6#%\"tG\"\"\"*&%\"aGF0--F*6#F-6#F/F0F0*&%\"bGF0-F-6#F/F0F0-%&deltaG6 #F/" }{TEXT -1 80 ", \351 possivel agora resolver a equa\347\343o com \+ termo n\343o homog\352neo qualquer, isto e: " }{XPPEDIT 256 0 "`@@`(D, 2)(y)(t)+a*D(y)(t)+b*y(t) = h(t);" "6#/,(---%#@@G6$%\"DG\"\"#6#%\"yG6# %\"tG\"\"\"*&%\"aGF0--F*6#F-6#F/F0F0*&%\"bGF0-F-6#F/F0F0-%\"hG6#F/" } {TEXT -1 13 " por meio de " }{TEXT 309 10 "convolu\347\343o" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "A convolu\347\343o f *g de duas fun\347\365es f(t) e g(t) \351 def inido pela integral" }}{PARA 256 "" 0 "" {TEXT -1 10 "(f*g)(t) =" } {XPPEDIT 18 0 "int(f(r)*g(t-r),r = 0 .. t)" "6#-%$intG6$*&-%\"fG6#%\"r G\"\"\"-%\"gG6#,&%\"tGF+F*!\"\"F+/F*;\"\"!F0" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 165 "N\343o confunda o asterisco * aqui com o s\355 mbolo usado por Maple para indicar a multiplica\347\343o. A propriedad e principal da convolu\347\343o \351 que a transformada de Laplace de \+ " }{TEXT 310 3 "f*g" }{TEXT -1 3 " \351 " }{TEXT 311 8 "F(s)G(s)" } {TEXT -1 7 ", onde " }{TEXT 312 5 "F(s) " }{TEXT -1 31 "\351 a transfo rmade de Laplace de " }{TEXT 313 4 "f(t)" }{TEXT -1 4 ", e " }{TEXT 314 4 "G(s)" }{TEXT -1 4 " de " }{TEXT 315 5 "g(t)." }}{PARA 0 "" 0 " " {TEXT -1 29 "Como a solu\347\343o particular de " }{XPPEDIT 256 0 "` @@`(D,2)(y)(t)+a*D(y)(t)+b*y(t) = h(t)" "6#/,(---%#@@G6$%\"DG\"\"#6#% \"yG6#%\"tG\"\"\"*&%\"aGF0--F*6#F-6#F/F0F0*&%\"bGF0-F-6#F/F0F0-%\"hG6# F/" }{TEXT -1 62 " com as condi\347\365es y(0)=0, y'(0)=0 tem transfor mada de Laplace " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "H (s)/(s^2+a*s+b);" "6#*&-%\"HG6#%\"sG\"\"\",(*$F'\"\"#F(*&%\"aGF(F'F(F( %\"bGF(!\"\"" }{TEXT -1 5 " = " }{XPPEDIT 18 0 "1/(s^2+a*s+b);" "6#* &\"\"\"F$,(*$%\"sG\"\"#F$*&%\"aGF$F'F$F$%\"bGF$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "H(s);" "6#-%\"HG6#%\"sG" }}{PARA 0 "" 0 "" {TEXT -1 116 "A transformada de Laplace inverse ent\343o seria a convolu\347 \343o da transformada de Laplace inversa do primeiro fator com " } {TEXT 316 4 "h(t)" }{TEXT -1 8 ". Isto \351" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 317 14 "y(t)=K(t)*h(t)" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 145 "O Maple n\343o possui comando pr \363prio para a convolu\347\343o e portanto devomos defin\355-la por u m procedimento usando a linguagem de programa\347\343o do Maple." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "conv:=proc(f,g) local t,r;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "int(apply(f,r)*apply(g,t-r),r=0.. t);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "end proc:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Vamos ver algumas convolu\347\365es. Primeiro a g en\351rica:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "conv(f,g);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "E agora algumas fun\347\365es c oncretas:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 319 6 "f(t)=t" }{TEXT -1 3 " e " }{TEXT 318 11 "g(t)=sin(t)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "conv(t->t,sin);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 320 11 "f(t)=sin(t)" }{TEXT -1 3 " e " }{TEXT 321 11 "g(t)=sin(t)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "conv(sin,sin);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 322 11 "f(t)=sin(t)" }{TEXT -1 3 " e " }{TEXT 323 11 "g(t)=exp(t)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "conv (sin,exp);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 228 "Vamos agora ilustr ar o uso da convolu\347\343o resolvendo algumas equa\347\365es n\343o \+ homog\352neas calculando a solu\347\343o fundamental depois calculando a convolu\347\343o desta com o termo n\343o homog\352neo. Escrevemos \+ um procedimento que reune as etapas." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "solnh:=proc(a,b,h) local K,U, eq:" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 37 "printf(\"%s\",\"A equa\347\343o diferencial\"):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "eq:=(D@@2)(y)(t)+a*D(y)(t)+b*y(t)=h (t): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "print(eq): " }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 26 "K:=x->subs(t=x,solf(a,b)):" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 42 "printf(\"%s\", \"A solu\347\343o fundamentl K(t)\") :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "print(K(t)): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "printf(\"%s\",\"A convolu\347\343o de K(t) com o termo n\343o homog\352neo\"):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "U :=conv(K,h):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "print(U): " }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "printf(\"%s\", \"A solu \347\343o da e.d.o. com y(0)=0,y'(0)=0\"):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "print(subs(dsolve(\{eq,y(0)=0,D(y)(0)=0\},y(t)),y(t)) ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "end proc:" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 180 "Nos exempl os embaixo ser\343o exibidos a equa\347\343o diferencial, a solu\347 \343o fundamental, a convolu\347\343o da mesma com o termo n\343o homo g\352neo, e a solu\347\343o particular com as condi\347\365es iniciais " }{TEXT 324 6 "y(0)=0" }{TEXT -1 3 " e " }{TEXT 325 7 "y'(0)=0" } {TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "solnh(0,4, t->sin(2*t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "solnh(1,17 /4,t->exp(-t)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "solnh(0, 4,t->sin(2*t)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "solnh(0, 1,t->t^2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "solnh(3,2,t-> t^2*exp(-2*t)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {MARK "76 1 0" 34 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }