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}}{PARA 0 "" 0 "" {TEXT -1 81 "Nesta woksheet est\341 todo o material visto em aula sobr e transformada de Laplace. " }}{PARA 0 "" 0 "" {TEXT -1 57 "Use-o como uma revis\343o, mas n\343o esque\347a de lapis e papel." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "Transforma da de Laplace " }}{EXCHG {PARA 0 "" 0 "" {TEXT 315 48 "Teorema (Exist encia da Transformada de Laplace)" }{TEXT 32 7 " Se " }{XPPEDIT 32 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT 32 26 " \351 de ordem exponencial, " }}{PARA 0 "" 0 "" {TEXT 32 36 "ent\343o sua transformada de Laplace " }{XPPEDIT 32 0 "Lf(t) = F(s)" "6#/-%#LfG6#%\"tG-%\"FG6#%\"sG" } {TEXT 32 13 " \351 dada por" }}{PARA 0 "" 0 "" {TEXT 32 3 " " } {XPPEDIT 32 0 "F(s) = int(f(t)*exp(-s*t),t = 0 .. infinity);" "6#/-%\" FG6#%\"sG-%$intG6$*&-%\"fG6#%\"tG\"\"\"-%$expG6#,$*&F'F0F/F0!\"\"F0/F/ ;\"\"!%)infinityG" }{TEXT 32 6 " . \n" }}{PARA 0 "" 0 "" {TEXT 32 23 "A integral definindo " }{XPPEDIT 32 0 "F(s)" "6#-%\"FG6#%\"sG" } {TEXT 32 22 " existe nos pontos " }{XPPEDIT 32 0 "tau < s" "6#2%$ta uG%\"sG" }{TEXT 32 4 " . \n" }}{PARA 0 "" 0 "" {TEXT 316 53 "Teorema \+ (Linearidade da Transformada de Laplace )" }{TEXT 32 10 " Sejam \+ " }{XPPEDIT 32 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT 32 5 " e " } {XPPEDIT 32 0 "g(t)" "6#-%\"gG6#%\"tG" }{TEXT 32 45 " \ntendo transfo rmada de Laplace dadas por " }{XPPEDIT 32 0 "F(s)" "6#-%\"FG6#%\"sG " }{TEXT 32 6 " e " }{XPPEDIT 32 0 "G(s)" "6#-%\"GG6#%\"sG" }{TEXT 32 27 " , respectivamente. Se " }{XPPEDIT 32 0 "a" "6#%\"aG" } {TEXT 32 6 " e " }{XPPEDIT 32 0 "b" "6#%\"bG" }{TEXT 32 25 " s\343 o constantes, ent\343o " }{XPPEDIT 32 0 "L(a*f(t) + b*g(t)) =a*F(s) + b*G(s)" "6#/-%\"LG6#,&*&%\"aG\"\"\"-%\"fG6#%\"tGF*F**&%\"bGF*-%\"gG6# F.F*F*,&*&F)F*-%\"FG6#%\"sGF*F**&F0F*-%\"GG6#F9F*F*" }{TEXT 32 4 " . \+ \n" }}{PARA 0 "" 0 "" {TEXT 317 47 "Teorema (Unicidade da Transformad a de Laplace)" }{TEXT 32 10 " Sejam " }{XPPEDIT 32 0 "f(t)" "6#-%\" fG6#%\"tG" }{TEXT 32 5 " e " }{XPPEDIT 32 0 "g(t)" "6#-%\"gG6#%\"tG " }{TEXT 32 45 " \ntendo Transformada de Laplace dadas por " } {XPPEDIT 32 0 "F(s)" "6#-%\"FG6#%\"sG" }{TEXT 32 6 " e " }{XPPEDIT 32 0 "G(s)" "6#-%\"GG6#%\"sG" }{TEXT 32 27 " , respectivamente. Se \+ " }{XPPEDIT 32 0 "F(s) = G(s)" "6#/-%\"FG6#%\"sG-%\"GG6#F'" }{TEXT 32 11 " ent\343o " }{XPPEDIT 32 0 "f(t) = g(t)" "6#/-%\"fG6#%\"tG- %\"gG6#F'" }{TEXT 256 3 " .\n" }}{PARA 0 "" 0 "" {TEXT 269 115 "Para t rabalhar com Transformada de Laplace no Maple, voc\352 precisa carreg ar os procedimentos \"Laplace transform\" ." }}{PARA 0 "" 0 "" {TEXT -1 14 "Fa\347a isto com " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(in ttrans):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 318 10 "\nExemplo 1" }{TEXT 324 1 " " }{TEXT 314 62 "Determine a transformada de Laplace da fun \347\343o degrau unit\341rio." }}{PARA 263 "" 0 "" {XPPEDIT 18 0 "f(t) = 1" "6#/-%\"fG6#%\"tG\"\"\"" }{TEXT 271 11 " se " }{XPPEDIT 18 0 "0 <=t" "6#1\"\"!%\"tG" }{TEXT 274 3 " < " }{XPPEDIT 18 0 "c" "6# %\"cG" }{TEXT 275 3 " ,\n" }{XPPEDIT 18 0 "f(t) = 0" "6#/-%\"fG6#%\"tG \"\"!" }{TEXT 273 12 " se " }{XPPEDIT 18 0 "c < t" "6#2%\"cG% \"tG" }{TEXT 276 2 " ." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 301 "c:='c': \+ f:='f': F:='F': g:='g': s:='s': t:='t': T:='T':\nf0 := t -> 1:\ng := t -> subs(T=t, int(f0(T)*exp(-s*T),T)):\nF := t -> subs(T=t, int(f0(T)* exp(-s*T),T=0..c)):\n`For 0 <= t <= c, f(t) ` = f0(t);\nInt(f(t)*exp( -s*t),t) = g(t);\n`F(s) = `, Int(f(t)*exp(-s*t),t=0..c) = F(s);\n`F(s) ` = simplify(F(s));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Veja o gr \341fico de f." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "f:=x -> p iecewise(x>0,1);#tomei c=0 aqui" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plo t(f(x),x=-2..4,y=0..2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 319 9 "Exempl o 2" }{TEXT 256 2 " " }{TEXT 293 1 " " }{TEXT -1 41 " Determine a tra nsformada de Laplace de " }{TEXT 295 1 " " }{XPPEDIT 18 0 "f(t) = e^( a*t)" "6#/-%\"fG6#%\"tG)%\"eG*&%\"aG\"\"\"F'F," }{TEXT 277 3 " .\n" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 338 "a:='a': f:='f': F:='F': g:='g': s: ='s': t:='t': T:='T':\nf0 := t -> exp(a*t):\n`f(t) ` = f0(t);\ng := pr oc(t,S)\n simplify(subs(T=t,int(f0(T)*exp(-S*T),T)))\nend:\nInt(f(t)* exp(-s*t),t) = g(t,s);\n`F(s) ` = subs(T=t, int(f0(T)*exp(-s*T),T=0..i nfinity));\n`F(s) ` = simplify(g(infinity,s) - g(0,s));\nF := s -> - s ubs(S=s, g(0,S)):\n`F(s) ` = F(s);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "`f(t) ` = exp(a*t);\n`F(s ) ` = laplace(exp(a*t), t, s);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 266 1 "\n" }{TEXT 320 9 "Exemplo 3" }{TEXT 256 1 " " }{TEXT -1 41 " Determin e a transformada de Laplace de " }{XPPEDIT 18 0 "f(t) = sinh(a*t)" "6 #/-%\"fG6#%\"tG-%%sinhG6#*&%\"aG\"\"\"F'F-" }{TEXT -1 11 " . \nComo \+ " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 5 " = " } {XPPEDIT 18 0 "sinh(at)" "6#-%%sinhG6#%#atG" }{TEXT -1 5 " = " } {XPPEDIT 18 0 "(exp(a*t) - exp(-a*t))/2" "6#*&,&-%$expG6#*&%\"aG\"\" \"%\"tGF*F*-F&6#,$*&F)F*F+F*!\"\"F0F*\"\"#F0" }{TEXT -1 17 " , usamos que " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "L[1](exp(a*t)) = 1/(s - a)" "6#/-&%\"LG6#\"\"\"6#-%$expG6#*&%\"aG\"\"\"%\"tGF/*&\"\"\"F/,&%\"sGF/F .!\"\"F5" }{TEXT -1 7 " e " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "L[2]( exp(-a*t)) = 1/(s + a)" "6#/-&%\"LG6#\"\"#6#-%$expG6#,$*&%\"aG\"\"\"% \"tGF0!\"\"*&\"\"\"F0,&%\"sGF0F/F0F2" }{TEXT 284 2 " ." }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 264 "L:='L':\n`f(t) ` = sinh(a*t);\n`f(t) ` = (exp (a*t)-exp(-a*t))/2;\nL1 :=laplace( exp(a*t), t, s):\nL2 :=laplace(exp( -a*t), t, s):\nL(exp(a*t)) = L1;\nL(exp(-a*t)) = L2; ` `;\n`F(s) ` = ( L(exp(a*t)) - L(exp(-a*t)))/2;\n`F(s) ` = (L1 - L2)/2;\n`F(s) ` = simp lify((L1 - L2)/2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Podemos ver ificar este resultado usando as rotinas do Maple." }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 56 "`f(t) ` = sinh(a*t);\n`F(s) ` = laplace(sinh(a*t), \+ t, s);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 1 "\n" }{TEXT 321 9 "Exemp lo 4" }{TEXT -1 43 " Determine a trasnformada de Laplace de " } {XPPEDIT 18 0 "f(t) = t" "6#/-%\"fG6#%\"tGF'" }{TEXT 286 3 " .\n" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 331 "a:='a': f:='f': F:='F': g:='g': s: ='s': t:='t': T:='T':\nf0 := t -> t:\n`f(t) ` = f0(t);\ng := proc(t,S) \n simplify(subs(T=t,int(f0(T)*exp(-S*T),T)))\nend:\nInt(f(t)*exp(-s* t),t) = g(t,s);\n`F(s) ` = subs(T=t, int(f0(T)*exp(-s*T),T=0..infinity ));\n`F(s) ` = simplify(g(infinity,s) - g(0,s));\nF := s -> - subs(S=s , g(0,S)):\n`F(s) ` = F(s);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "Po demos verificar este resultado usando as rotinas do pacote Transformad a de Laplace." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "`f(t) ` = t;\n`F( s) ` = laplace(t, t, s);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 289 1 "\n" } {TEXT 322 9 "Exemplo 5" }{TEXT 256 1 " " }{TEXT -1 43 " Determine a t ransformada de Laplace de " }{XPPEDIT 18 0 "f(t) = cos(b*t)" "6#/-% \"fG6#%\"tG-%$cosG6#*&%\"bG\"\"\"F'F-" }{TEXT 291 3 " .\n" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 338 "a:='a': f:='f': F:='F': g:='g': s:='s': t:= 't': T:='T':\nf0 := t -> cos(b*t):\n`f(t) ` = f0(t);\ng := proc(t,S)\n simplify(subs(T=t,int(f0(T)*exp(-S*T),T)))\nend:\nInt(f(t)*exp(-s*t) ,t) = g(t,s);\n`F(s) ` = subs(T=t, int(f0(T)*exp(-s*T),T=0..infinity)) ;\n`F(s) ` = simplify(g(infinity,s) - g(0,s));\nF := s -> - subs(S=s, \+ g(0,S)):\n`F(s) ` = F(s);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Verf ique este resultado usando as rotinas do Maple." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "`f(t) ` = cos(b*t); \n`F(s) ` = laplace(cos(b*t), t, s);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 323 9 "Exemplo 6" }{TEXT 256 2 " " }{TEXT 294 2 " " }{TEXT -1 38 "De termine a transformada inversa de " }{XPPEDIT 18 0 "F(s) = (3*s + 6) /(s^2 + 9)" "6#/-%\"FG6#%\"sG*&,&*&\"\"$\"\"\"F'F,F,\"\"'F,F,,&*$F'\" \"#F,\"\"*F,!\"\"" }{TEXT -1 2 " ." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "f:='f': F:='F': s:='s': t:='t':\nF0 := s -> (3*s + 6)/(s^2 + 9): \n`F(s) ` = F0(s);\n`F(s) ` = expand(F0(s));" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 17 "A transformada " }{XPPEDIT 18 0 "F(s)" "6#-%\"FG6#%\" sG" }{TEXT -1 27 " \351 uma combina\347\343o linear ." }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 113 "F1 := s/(s^2 + 9):\nF2 := 3/(s^2 + 9):\nF[1]( s) = F1;\nF[2](s) = F2;\n`F(s) = `, 3*F[1](s) + 2*F[2](s) = 3*F1 + 2*F 2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "A inversa de " }{XPPEDIT 18 0 "F[1](s)" "6#-&%\"FG6#\"\"\"6#%\"sG" }{TEXT -1 6 " \351 " } {XPPEDIT 18 0 "cos(3*t)" "6#-%$cosG6#*&\"\"$\"\"\"%\"tGF(" }{TEXT -1 19 " e a inversa de " }{XPPEDIT 18 0 "F[2](s)" "6#-&%\"FG6#\"\"#6#% \"sG" }{TEXT -1 6 " \351 " }{XPPEDIT 18 0 "sin(3*t)" "6#-%$sinG6#*& \"\"$\"\"\"%\"tGF(" }{TEXT -1 1 " " }{TEXT 287 3 ". " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "f1 := invlaplace(F1, s, t):\nf2 := invlaplace(F 2, s, t):\nf[1](t) , ` = L^-1 (F1(s)) ` = f1;\nf[2](t) , ` = L^-1 (F2( s)) ` = f2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Portanto " } {TEXT 288 2 " " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 5 " = " }{XPPEDIT 18 0 "3*f[1](t) + 2*f[2](t)" "6#,&*&\"\"$\"\"\"-&%\"f G6#\"\"\"6#%\"tGF&F&*&\"\"#F&-&F)6#\"\"#6#F-F&F&" }{TEXT -1 5 " = " }{XPPEDIT 18 0 "3*cos(3*t) + 2*sin(3*t)" "6#,&*&\"\"$\"\"\"-%$cosG6#*& \"\"$F&%\"tGF&F&F&*&\"\"#F&-%$sinG6#*&\"\"$F&F,F&F&F&" }{TEXT 292 2 " \+ ." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "`f(t) = `, 3*f[1](t) + 2*f[2]( t) = 3*f1 + 2*f2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Podemos veri ficar isto usando os procedimentos do Maple." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "`F(s) ` = (3*s + 6)/(s^2 + 9);\n`f(t) ` = invlaplace( (3*s + 6)/(s^2 + 9), s, t);" }}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 48 "Transformada de Laplace de derivadas e in tegrais" }}{PARA 0 "" 0 "" {TEXT 325 30 "Teorema (Derivada de f(t)) " }{TEXT 32 7 " Seja " }{XPPEDIT 32 0 "f(t)" "6#-%\"fG6#%\"tG" } {TEXT 32 5 " e " }{XPPEDIT 32 0 "`f '(t)`" "6#%'f~'(t)G" }{TEXT 32 17 " cont\355nuas em " }}{PARA 0 "" 0 "" {XPPEDIT 32 0 "t>=0" "6#1 \"\"!%\"tG" }{TEXT 32 39 " , e de ordem exponencial. Ent\343o, " }{XPPEDIT 32 0 "L(`f '(t)`) = s*F(s) - f(0)" "6#/-%\"LG6#%'f~'(t)G,&*& %\"sG\"\"\"-%\"FG6#F*F+F+-%\"fG6#\"\"!!\"\"" }{TEXT 32 11 " , onde \+ " }{XPPEDIT 32 0 "L(`f(t)`) = F(s)" "6#/-%\"LG6#%%f(t)G-%\"FG6#%\"sG" }{TEXT 32 3 " .\n" }}{PARA 0 "" 0 "" {TEXT 326 30 "Teorema (Integra \347\343o de f(t))" }{TEXT 32 10 " Seja " }{XPPEDIT 32 0 "f(t)" " 6#-%\"fG6#%\"tG" }{TEXT 32 5 " e " }{XPPEDIT 32 0 "`f '(t)`" "6#%'f~ '(t)G" }{TEXT 32 19 " cont\355nuas para " }{XPPEDIT 32 0 "0<=t" "6# 1\"\"!%\"tG" }{TEXT 32 41 " , \ne de ordem exponencial . Ent\343o, \+ " }{XPPEDIT 32 0 "L(Int(f (t), t)) = F(s)/s" "6#/-%\"LG6#-%$IntG6$-% \"fG6#%\"tGF-*&-%\"FG6#%\"sG\"\"\"F2!\"\"" }{TEXT 32 10 " , onde " } {XPPEDIT 32 0 "L(f(t)) = F(s)" "6#/-%\"LG6#-%\"fG6#%\"tG-%\"FG6#%\"sG " }{TEXT 32 3 " .\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 56 "Vamos ver alguns exemplos de como usar estes resultados ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 32 71 "Carr egue os procedimentos para transformada de Laplace. Fa\347a isto com. \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(inttrans):" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 272 1 "\n" }{TEXT 327 9 "Exemplo 1" } {TEXT 256 8 " " }{TEXT 32 9 "Determine" }{TEXT 256 1 " " } {TEXT -1 1 " " }{XPPEDIT 18 0 "L(cos(t)^2)" "6#-%\"LG6#*$-%$cosG6#%\"t G\"\"#" }{TEXT 265 2 " ." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "f:='f' : F:='F': s:='s': t:='t': T:='T':\nf := t -> cos(t)^2:\nf1 := t -> sub s(T=t,diff(f(T),T)):\n`f(t) ` = f(t);\n`f(0) ` = f(0);\n`f '(t) ` = f1 (t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Note que " }{XPPEDIT 18 0 "sin(2*t)= 2*sin(t)*cos(t)" "6#/-%$sinG6#*&\"\"#\"\"\"%\"tGF)*(\" \"#F)-F%6#F*F)-%$cosG6#F*F)" }{TEXT -1 21 " e assim use que " } {XPPEDIT 18 0 "L(sin(2*t)) = 2/(s^2 + 4)" "6#/-%\"LG6#-%$sinG6#*&\"\"# \"\"\"%\"tGF,*&\"\"#F,,&*$%\"sG\"\"#F,\"\"%F,!\"\"" }{TEXT 267 2 " ." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "LDf := - 2/(s^2 + 4):\nLDf = `L( f '(t)) `;\neqn := LDf = s*F(s) - f(0): eqn;\nsol := solve(eqn, F(s)) :\n`Resolva para F(s).`;\n`F(s) ` = sol;\nsol := simplify(sol):\n`F(s ) ` = sol;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Verifique isto usan do as rotinas para Laplace" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "`f(t) ` = cos(t)^2;\n`F(s) ` = laplace(cos( t)^2, t, s);" }}}{EXCHG {PARA 264 "" 0 "" {TEXT 257 38 "Surpresa, o M aple N\303O pode calcular!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "\n " }{TEXT 328 10 "Exemplo 2 " }{TEXT 32 1 " " }{TEXT -1 39 " Use o teor ema acima apra determinar " }{XPPEDIT 18 0 "L(f(t))" "6#-%\"LG6#-%\" fG6#%\"tG" }{TEXT -1 7 " ,onde " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 330 4 "(a) " }{TEXT 329 2 " " }{XPPEDIT 18 0 "L (t^2)" "6#-%\"LG6#*$%\"tG\"\"#" }{TEXT -1 10 " .\nComo " }{XPPEDIT 18 0 "`f '(t)` = 2*t" "6#/%'f~'(t)G*&\"\"#\"\"\"%\"tGF'" }{TEXT -1 6 " e " }{XPPEDIT 18 0 "L(2*t) = 2/s^2" "6#/-%\"LG6#*&\"\"#\"\"\"%\"tG F)*&\"\"#F)*$%\"sG\"\"#!\"\"" }{TEXT -1 2 " ." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 206 "f:='f': F:='F': s:='s': t:='t':\nf := t -> t^2:\nf1 \+ := t -> subs(T=t,diff(f(T),T)):\n`f(t) ` = f(t);\n`f '(t) ` = f1(t);\n LDf := laplace(f1(t), t, s):\n`L(f '(t)) ` = LDf;\nLf := LDf/s:\n`F(s) = L(f '(t))/s ` = Lf;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Verifiq ue isto com as rotinas do Maple para Laplace." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "`f(t) ` = t^2;\n`F(s) ` = laplace(t^2, t, s);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 332 4 "(b) " }{TEXT 331 3 " " } {XPPEDIT 18 0 "L(t^3)" "6#-%\"LG6#*$%\"tG\"\"$" }{TEXT 278 3 " .\n" }} {PARA 0 "" 0 "" {TEXT 259 7 "Como " }{XPPEDIT 18 0 "`f '(t)` = 3*t^2 " "6#/%'f~'(t)G*&\"\"$\"\"\"*$%\"tG\"\"#F'" }{TEXT 270 6 " e " } {XPPEDIT 18 0 "L(3*t^2) = 6/s^3" "6#/-%\"LG6#*&\"\"$\"\"\"*$%\"tG\"\"# F)*&\"\"'F)*$%\"sG\"\"$!\"\"" }{TEXT 279 2 " ." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 206 "f:='f': F:='F': s:='s': t:='t':\nf := t -> t^3:\nf1 \+ := t -> subs(T=t,diff(f(T),T)):\n`f(t) ` = f(t);\n`f '(t) ` = f1(t);\n LDf := laplace(f1(t), t, s):\n`L(f '(t)) ` = LDf;\nLf := LDf/s:\n`F(s) = L(f '(t))/s ` = Lf;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Verifiq ue isto com as rotinas do Maple." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "`f(t) ` = t^3;\n`F(s) ` = laplace(t^3, t, s);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 280 1 "\n" }{TEXT 334 11 "Exemplo 3 " }{TEXT 333 1 " " } {TEXT -1 47 " Resolver o seguinte problema de valor inicial " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "`y''(t) + y(t) = 0`" "6#%2y''(t)~+~y(t)~=~0G " }{TEXT -1 10 " com " }{XPPEDIT 18 0 "y(0) = 2" "6#/-%\"yG6#\"\" !\"\"#" }{TEXT -1 6 " e " }{XPPEDIT 18 0 "`y'(0)` = 3" "6#/%&y'(0)G \"\"$" }{TEXT -1 2 " ." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 246 "s:='s': \+ t:='t': Y:='Y': Ys:='Ys':\ny0 := 2:\ny1 := 3:\nF := 0:\n`y''(t) + y( t) = 0`;\n`y(0) ` = y0,` y'(0) ` = y1;\neqn := s^2*Y(s) - s*y0 - \+ y1 + Y(s) = F: eqn;\nsol := simplify(solve(subs(s=S,eqn),Y(S))):\nY := s -> subs(S=s,sol):\n`Y(s) ` = Y(s);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Usando que " }{TEXT 281 1 " " }{XPPEDIT 18 0 "L(cos(t)) = s/(s^2 + 1)" "6#/-%\"LG6#-%$cosG6#% \"tG*&%\"sG\"\"\",&*$F,\"\"#F-\"\"\"F-!\"\"" }{TEXT 290 8 " e " } {XPPEDIT 18 0 "L(sin(t)) = 1/(s^2 + 1)" "6#/-%\"LG6#-%$sinG6#%\"tG*&\" \"\"\"\"\",&*$%\"sG\"\"#F-\"\"\"F-!\"\"" }{TEXT 282 1 "\n" }{TEXT -1 76 "vamos determinar a solu\347\343o que \351 uma combina\347\343o lin ear de cos(t) e sin(t) ." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "F1 := s/(s^2+1):\nF2 := 1/(s^2+1):\n`Y(s) ` = 2*F1 + 3*F2;\nf1 := invlaplac e(F1, s , t):\nf2 := invlaplace(F2, s , t):\n`f(t) ` = 2*f1 + 3*f2;" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "Podemos usar o Maple para determ inar diretamente a inversa." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "`Y( s) ` = Y(s);\n`f(t) ` = invlaplace(Y(s), s , t);" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 81 "Podemos usar os procedimentos do Maple para EDO pa ra obter a solu\347\343o diretamente." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 166 "y:='y': t:='t':\nODE := diff(y(t),t$2)+y(t) = 0:\nICs := \{y(0 )=2, D(y)(0)=3\}:\n`D. E. ` = ODE;\n`I. C.'s ` = ICs;\ndsolve(OD E, y(t));\ndsolve(\{ODE\} union ICs, y(t));" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 283 1 "\n" }{TEXT 335 11 "Exemplo 4 " }{TEXT -1 38 " Resolva o problema de valor inicial " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "`y''(t ) + y'(t) - 2y(t) = 0`" "6#%;y''(t)~+~y'(t)~-~2y(t)~=~0G" }{TEXT -1 10 " com " }{XPPEDIT 18 0 "y(0) = 1" "6#/-%\"yG6#\"\"!\"\"\"" } {TEXT -1 5 " e " }{XPPEDIT 18 0 "`y'(0)` = 4" "6#/%&y'(0)G\"\"%" } {TEXT -1 2 " ." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 308 "s:='s': S:='S': \+ t:='t': Y:='Y':\ny0 := 1:\ny1 := 4:\nF := 0:\n`y''(t) + y'(t) - 2y (t) = 0`;\n`y(0) ` = y0, ` y'(0) ` = y1;\neqn := s^2*Y(s) - s*y0 - y1 + s*Y(s) - y0 - 2*Y(s) = F:\neqn;\nsol := simplify(solve(subs(s=S, eqn),Y(S))):\nY := s -> subs(S=s,sol):\n`Y(s) ` = Y(s);\n`Y(s) ` = con vert(Y(s), parfrac, s);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Usando que " }{XPPEDIT 18 0 "L(exp(-2*t)) = 1/(s + 2)" "6#/-%\"LG6#-%$expG 6#,$*&\"\"#\"\"\"%\"tGF-!\"\"*&\"\"\"F-,&%\"sGF-\"\"#F-F/" }{TEXT -1 9 " e " }{XPPEDIT 18 0 "L(exp(t)) = 1/(s - 1)" "6#/-%\"LG6#-%$ex pG6#%\"tG*&\"\"\"\"\"\",&%\"sGF-\"\"\"!\"\"F1" }{TEXT 285 1 "\n" } {TEXT -1 31 "a solu\347\343o \351 combina\347\343o linear ." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "F1 := 1/(s+2):\nF2 := 1/(s-1):\n`Y(s) ` \+ = - F1 + 2*F2;\nf1 := invlaplace(F1, s , t):\nf2 := invlaplace(F2, s , t):\n`f(t) ` = - f1 + 2*f2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "P odemos usar a inversa de Laplace diretamente." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "`Y(s) ` = Y(s);\n`f(t) ` = invlaplace(Y(s), s , t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Podemos verificar isto com o Ma ple." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "y:='y': t:='t':\nODE := d iff(y(t),t$2)+diff(y(t),t)-2*y(t) = 0:\nICs := \{y(0)=1, D(y)(0)=4\}: \n`D. E. ` = ODE;\n`I. C.'s ` = ICs;\ndsolve(ODE, y(t));\ndsolve (\{ODE\} union ICs, y(t));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 346 1 "\n" }{TEXT 348 11 "Exemplo 5 " }{TEXT -1 38 " Resolva o problema de valor inicial " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "`y''(t) -3 y'(t) + 2y(t) = 0`" "6#% " 0 "" {MPLTEXT 1 0 311 "s:='s': S:='S': t:='t': Y:='Y':\ny0 := -1:\ny1 := 4:\nF := 0: \n`y''(t) -3y'(t) + 2y(t) = 0`;\n`y(0) ` = y0, ` y'(0) ` = y1;\n eqn := s^2*Y(s) - s*y0 - y1 -3*(s*Y(s) - y0) + 2*Y(s) = F:\neqn;\nsol := simplify(solve(subs(s=S,eqn),Y(S))):\nY := s -> subs(S=s,sol):\n`Y (s) ` = Y(s);\n`Y(s) ` = convert(Y(s), parfrac, s);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Usando que " }{XPPEDIT 18 0 "L(exp(t)) = 1/(s -1)" "6#/-%\"LG6#-%$expG6#%\"tG*&\"\"\"\"\"\",&%\"sGF-\"\"\"!\"\"F1" }{TEXT -1 9 " e " }{XPPEDIT 18 0 "L(exp(2*t)) = 1/(s -2)" "6#/-% \"LG6#-%$expG6#*&\"\"#\"\"\"%\"tGF,*&\"\"\"F,,&%\"sGF,\"\"#!\"\"F3" } {TEXT 347 1 "\n" }{TEXT -1 31 "a solu\347\343o \351 combina\347\343o l inear ." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "F1 := 1/(s-2):\nF2 := 1 /(s-1):\n`Y(s) ` = 5*F1 - 6*F2;\nf1 := invlaplace(F1, s , t):\nf2 := \+ invlaplace(F2, s , t):\n`f(t) ` = 5* f1 -6*f2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Podemos usar a inversa de Laplace diretamente." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "`Y(s) ` = Y(s);\n`f(t) ` = invlapla ce(Y(s), s , t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Podemos verif icar isto com o Maple." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "y:='y': \+ t:='t':\nODE := diff(y(t),t$2)-3*diff(y(t),t)+2*y(t) = 0:\nICs := \{y (0)=-1, D(y)(0)=4\}:\n`D. E. ` = ODE;\n`I. C.'s ` = ICs;\ndsolve (ODE, y(t));\ndsolve(\{ODE\} union ICs, y(t));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "Teoremas de Desl ocamento" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 336 38 "Teorema (deslocamento na vari\341vel s)" }{TEXT 32 7 " Se " } {XPPEDIT 32 0 "F(s)" "6#-%\"FG6#%\"sG" }{TEXT 32 35 " \351 a transfor mada de Laplace de " }{XPPEDIT 32 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT 32 11 " ,\nent\343o " }{XPPEDIT 32 0 "L(exp(a*t)*f(t)) = F(s - a)" " 6#/-%\"LG6#*&-%$expG6#*&%\"aG\"\"\"%\"tGF-F--%\"fG6#F.F--%\"FG6#,&%\"s GF-F,!\"\"" }{TEXT 32 3 " .\n" }}{PARA 0 "" 0 "" {TEXT 337 39 "Teorema (deslocamento na vari\341vel t) " }{TEXT 32 5 " Se " }{XPPEDIT 32 0 "F(s)" "6#-%\"FG6#%\"sG" }{TEXT 32 37 " \351 a transformada de Lap lace de " }{XPPEDIT 32 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT 32 7 " ,\n e " }{XPPEDIT 32 0 "0 <= a" "6#1\"\"!%\"aG" }{TEXT 32 12 " , ent \343o " }{XPPEDIT 32 0 "L(U[a](t)*f(t-a)) = exp(-a*s)*F(s)" "6#/-%\" LG6#*&-&%\"UG6#%\"aG6#%\"tG\"\"\"-%\"fG6#,&F.F/F,!\"\"F/*&-%$expG6#,$* &F,F/%\"sGF/F4F/-%\"FG6#F;F/" }{TEXT 32 3 " .\n" }}{PARA 0 "" 0 "" {TEXT 32 63 "Carregue o pacote de transforma\347\365es para trabalhar \+ com o Maple." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(inttran s):\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 302 1 "\n" }{TEXT 339 11 "Exemp lo 1 " }{TEXT 338 2 " " }{TEXT 258 10 "Calcule " }{XPPEDIT 18 0 "L (t^n*exp(a*t))" "6#-%\"LG6#*&)%\"tG%\"nG\"\"\"-%$expG6#*&%\"aGF*F(F*F* " }{TEXT 303 1 " " }{TEXT 260 2 "\n " }{TEXT 261 13 "Usando que " } {XPPEDIT 18 0 "L(t^n) = n!/s^(n+1)" "6#/-%\"LG6#)%\"tG%\"nG*&-%*factor ialG6#F)\"\"\")%\"sG,&F)F.\"\"\"F.!\"\"" }{TEXT 304 28 " . e fazemos o deslocamento." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 222 "a:='a': f:='f': \+ F:='F': n:='n': s:='s': t:='t': \nf := t -> t^n:\nF := s -> n!/s^(n+1 ):\n`formulas dadas:`;\n`f(t) ` = f(t);\n`F(s) ` = F(s);\n`deslocament o na variavel s para obter:`;\n`f(t) ` = f(t)*exp(a*t);\n`F(s) ` = F(s -a);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 297 35 "Podemos verificar isto c om o Maple." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 268 "assume(A,positive); \nassume(N,positive);\n`Por exemplo, comece com:`;\n`f(t) ` = t^n;\nL \+ := laplace(t^N, t, s):\n`F(s) ` = subs(N='n',L);\n`Deslocamento na var i\341vel s para obter:`;\n`f(t) ` = t^n*exp(a*t);\nLS := laplace(t^N*e xp(A*t), t, s):\n`F(s) ` = subs(\{A='a',N='n'\},LS);;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 341 9 "Exemplo 2 " }{TEXT 32 2 " " }{TEXT 340 1 " " }{TEXT 262 15 "Resolva o PVI " }} {PARA 0 "" 0 "" {XPPEDIT 18 0 "`y''(t) + y(t)` = U[pi](t)" "6#/%.y''(t )~+~y(t)G-&%\"UG6#%#piG6#%\"tG" }{TEXT 305 10 " com " }{XPPEDIT 18 0 "y(0) = 0" "6#/-%\"yG6#\"\"!F'" }{TEXT 306 5 " e " }{XPPEDIT 18 0 "`y'(0) = 0`" "6#%*y'(0)~=~0G" }{TEXT 307 1 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 385 "s:='s': S:='S': t:='t': Y:='Y': Y:='Y':\ny0 := 0: \ny1 := 0:\nF := laplace(Heaviside(t-Pi), t, s):\n`y''(t) + y(t) = \+ UPi(t)`;\n`y(0) ` = y0,` y'(0) ` = y1;\neqn := s^2*Y(s) - s*y0 - y 1 + Y(s) = F: eqn;\nsol := simplify(solve(subs(s=S,eqn),Y(S))):\nY := \+ s -> subs(S=s,sol):\n`Y(s) ` = Y(s);\n1/(s*(s^2+1)) = convert(1/(s*(s^ 2+1)), parfrac, s);\n`Y(s) ` = exp(-Pi*s)/s - exp(-Pi*s)*s/(s^2+1);" } }}{EXCHG {PARA 0 "" 0 "" {TEXT 298 10 "Use que " }{XPPEDIT 18 0 "L(U [pi](t)) = exp(-pi*s)/s" "6#/-%\"LG6#-&%\"UG6#%#piG6#%\"tG*&-%$expG6#, $*&F+\"\"\"%\"sGF4!\"\"F4F5F6" }{TEXT 299 9 " e " }{XPPEDIT 18 0 "L(U[pi](t)*cos(t - pi)) = exp(-pi*s)*s/(s^2 + 1)" "6#/-%\"LG6#*&-&% \"UG6#%#piG6#%\"tG\"\"\"-%$cosG6#,&F.F/F,!\"\"F/*(-%$expG6#,$*&F,F/%\" sGF/F4F/F;F/,&*$F;\"\"#F/\"\"\"F/F4" }{TEXT 300 3 " . " }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 143 "F1 := exp(-Pi*s)/s:\nF2 := exp(-Pi*s)*s/(s^2+ 1):\nf1 := invlaplace(F1, s , t):\nf2 := invlaplace(F2, s , t):\n`Y(s) ` = F1 - F2;\n`f(t) ` = f1 - f2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 301 38 "Podemos verificar isto usando o Maple." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 196 "ODE := diff(y(t),t$2)+y(t) = Heaviside(t-Pi):\nICs \+ := \{y(0)=0, D(y)(0)=0\}:\n`D. E. ` = ODE;\n`I. C.'s ` = ICs;\nd solve(ODE, y(t), method=laplace);\ndsolve(\{ODE\} union ICs, y(t), met hod=laplace);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 36 "Invertendo a Transformada de Laplace" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "with(inttrans):\nlaplace(t,t,s):" } }}{EXCHG {PARA 0 "" 0 "" {TEXT 342 9 "Exemplo 1" }{TEXT 256 2 " " } {TEXT 32 34 "Determine a transrformada inversa " }{TEXT 343 4 " de " } {XPPEDIT 18 0 "Y(s) = (s^3 - 4*s + 1)/(s*(s - 1)^3)" "6#/-%\"YG6#%\"sG *&,(*$F'\"\"$\"\"\"*&\"\"%F,F'F,!\"\"\"\"\"F,F,*&F'F,*$,&F'F,\"\"\"F/ \"\"$F,F/" }{TEXT 311 2 " ." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "s:=' s': Y:='Y':\nY := s ->(s^3 - 4*s + 1)/(s*(s-1)^3):\n`Y(s) ` = Y(s);" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "Vamos usar fra\347\365es parcia is para decompor a express\343o de Y(s) em fatores mais simples" } {TEXT 263 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "s:='s': Y:='Y ':\nY := s ->(s^3 - 4*s + 1)/(s*(s-1)^3):\n`Y(s) ` = Y(s);\n`Y(s) ` = \+ convert(Y(s),parfrac,s);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "Usamo s a tabela de transformada de Laplace para procurar a inversa e obtemo s a solu\347\343o: " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "f(t) = - 1 - t^2 *exp(t) + t*exp(t) + 2*exp(t)" "6#/-%\"fG6#%\"tG,*\"\"\"!\"\"*&F'\"\"# -%$expG6#F'\"\"\"F**&F'F0-F.6#F'F0F0*&\"\"#F0-F.6#F'F0F0" }{TEXT 312 2 " \n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Podemos usar o Maple para conferir." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "`F(s) ` = (s^3 - 4*s + 1)/(s*(s-1)^3);\n`f(t) ` = invlaplace((s^ 3 - 4*s + 1)/(s*(s-1)^3), s, t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 310 1 "\n" }{TEXT 344 9 "Exemplo 2" }{TEXT 32 37 " Determine a transforma da inversa de" }{TEXT -1 3 " " }{XPPEDIT 18 0 "F(s) = 5*s/((s^2 + 4) *(s^2+ 9))" "6#/-%\"FG6#%\"sG*(\"\"&\"\"\"F'F**&,&*$F'\"\"#F*\"\"%F*F* ,&*$F'\"\"#F*\"\"*F*F*!\"\"" }{TEXT -1 2 " ." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "s:='s': Y:='Y':\nY := s -> 5*s/((s^2+4)*(s^2+9)):\n`Y (s) ` = Y(s);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Podemos usar o M aple para converter em fra\347\365es parciais." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "s:='s': Y:='Y':\nY := s -> 5*s/((s^2+4)*(s^2+9)):\n`Y (s) ` = Y(s);\n`Y(s) ` = convert(Y(s),parfrac,s);" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 67 "Use a tabela de transformadad e Laplace para deter minar a inversa: " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "y(t) = cos(2*t) - \+ cos(3*t)" "6#/-%\"yG6#%\"tG,&-%$cosG6#*&\"\"#\"\"\"F'F.F.-F*6#*&\"\"$F .F'F.!\"\"" }{TEXT -1 3 " .\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 345 12 "Exemplo 3 " }{TEXT -1 38 "Determine a transformada inversa de " } {XPPEDIT 18 0 "Y(s) = (s^3+3*s^2-s+1)/(s*(s+1)^2*(s^2+1))" "6#/-%\"YG6 #%\"sG*&,**$F'\"\"$\"\"\"*&\"\"$F,*$F'\"\"#F,F,F'!\"\"\"\"\"F,F,*(F'F, *$,&F'F,\"\"\"F,\"\"#F,,&*$F'\"\"#F,\"\"\"F,F,F1" }{TEXT 313 3 " .\n" }{TEXT -1 25 "Em fra\347oes parciais temos" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "s:='s': Y:='Y':\nY := s ->(s^3+3*s^2-s+1)/(s*(s+1)^2 *(s^2+1)):\n`Y(s) ` = Y(s);\n`Y(s) ` = convert(Y(s),parfrac,s);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Usando uma tabela de transformada \+ de Laplace temos que a resposta \351:" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "y(t) = 1 - 2*t*exp(-t) - 2*exp(-t) + cos(t) + sin(t)" "6#/-%\"yG6#% \"tG,,\"\"\"\"\"\"*(\"\"#F*F'F*-%$expG6#,$F'!\"\"F*F1*&\"\"#F*-F.6#,$F 'F1F*F1-%$cosG6#F'F*-%$sinG6#F'F*" }{TEXT -1 2 " ." }{TEXT 264 1 "\n" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 15 "Exemplos de PVI" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(inttrans):" }}}{PARA 0 "" 0 " " {TEXT 352 11 "Exemplo 1 " }{TEXT -1 38 " Resolva o problema de val or inicial " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "`y''(t) -3 y'(t) + 2y(t) = 0`" "6#% " 0 "" {MPLTEXT 1 0 310 "s:='s': S:='S': t:='t': Y:= 'Y':\ny0 := 3:\ny1 := 4:\nF := 0:\n`y''(t) -3y'(t) + 2y(t) = 0`;\n `y(0) ` = y0, ` y'(0) ` = y1;\neqn := s^2*Y(s) - s*y0 - y1 -3*(s*Y( s) - y0) + 2*Y(s) = F:\neqn;\nsol := simplify(solve(subs(s=S,eqn),Y(S) )):\nY := s -> subs(S=s,sol):\n`Y(s) ` = Y(s);\n`Y(s) ` = convert(Y(s) , parfrac, s);" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Usando que " }{XPPEDIT 18 0 "L(exp(t)) = 1/(s -1) " "6#/-%\"LG6#-%$expG6#%\"tG*&\"\"\"\"\"\",&%\"sGF-\"\"\"!\"\"F1" } {TEXT -1 9 " e " }{XPPEDIT 18 0 "L(exp(2*t)) = 1/(s - 2)" "6#/-% \"LG6#-%$expG6#*&\"\"#\"\"\"%\"tGF,*&\"\"\"F,,&%\"sGF,\"\"#!\"\"F3" } {TEXT 351 1 "\n" }{TEXT -1 31 "a solu\347\343o \351 combina\347\343o l inear ." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "F1 := 1/(s-2):\nF2 := 1 /(s-1):\n`Y(s) ` = F1 + 2*F2;\nf1 := invlaplace(F1, s , t):\nf2 := in vlaplace(F2, s , t):\n`f(t) ` = f1 + 2*f2;" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 46 "Podemos usar a inversa de Laplace diretamente." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "`Y(s) ` = Y(s);\n`f(t) ` = invlapla ce(Y(s), s , t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Podemos verif icar isto com o Maple." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 183 "y:='y': \+ t:='t':\nODE := diff(y(t),t$2)-3*diff(y(t),t)+2*y(t) = 0:\nICs := \{y (0)=3, D(y)(0)=4\}:\n`D. E. ` = ODE;\n`I. C.'s ` = ICs;\ndsolve( ODE, y(t));\ndsolve(\{ODE\} union ICs, y(t));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 350 11 "Exemplo 2 " }{TEXT -1 38 " Resolva o problema de valor inicial \+ " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "`y''(t) -3 y'(t) + 2y(t) = 0`" "6#% " 0 "" {MPLTEXT 1 0 311 "s:='s': S:='S': t:='t': Y:= 'Y':\ny0 := -1:\ny1 := 4:\nF := 0:\n`y''(t) -3y'(t) + 2y(t) = 0`; \n`y(0) ` = y0, ` y'(0) ` = y1;\neqn := s^2*Y(s) - s*y0 - y1 -3*(s* Y(s) - y0) + 2*Y(s) = F:\neqn;\nsol := simplify(solve(subs(s=S,eqn),Y( S))):\nY := s -> subs(S=s,sol):\n`Y(s) ` = Y(s);\n`Y(s) ` = convert(Y( s), parfrac, s);" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " " 0 "" {TEXT -1 13 "Usando que " }{XPPEDIT 18 0 "L(exp(t)) = 1/(s -1 )" "6#/-%\"LG6#-%$expG6#%\"tG*&\"\"\"\"\"\",&%\"sGF-\"\"\"!\"\"F1" } {TEXT -1 9 " e " }{XPPEDIT 18 0 "L(exp(2*t)) = 1/(s - 2)" "6#/-% \"LG6#-%$expG6#*&\"\"#\"\"\"%\"tGF,*&\"\"\"F,,&%\"sGF,\"\"#!\"\"F3" } {TEXT 349 1 "\n" }{TEXT -1 31 "a solu\347\343o \351 combina\347\343o l inear ." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "F1 := 1/(s-2):\nF2 := 1 /(s-1):\n`Y(s) ` = -6* F1 + 5*F2;\nf1 := invlaplace(F1, s , t):\nf2 := invlaplace(F2, s , t):\n`f(t) ` = - 6*f1 + 5*f2;" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 46 "Podemos usar a inversa de Laplace diretamente." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "`Y(s) ` = Y(s);\n`f(t) ` = invlapla ce(Y(s), s , t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Podemos verif icar isto com o Maple." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "y:='y': \+ t:='t':\nODE := diff(y(t),t$2)-3*diff(y(t),t)+2*y(t) = 0:\nICs := \{y (0)=-1, D(y)(0)=4\}:\n`D. E. ` = ODE;\n`I. C.'s ` = ICs;\ndsolve (ODE, y(t));\ndsolve(\{ODE\} union ICs, y(t));" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(inttrans):" }}}{PARA 0 "" 0 "" {TEXT 354 11 "Exemplo 3 \+ " }{TEXT -1 38 " Resolva o problema de valor inicial " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "`y''(t) + y(t) = t`" "6#%3y''(t)~~+~y(t)~=~tG" } {TEXT -1 10 " com " }{XPPEDIT 18 0 "y(0) = -1" "6#/-%\"yG6#\"\"!, $\"\"\"!\"\"" }{TEXT -1 5 " e " }{XPPEDIT 18 0 "`y'(0)` = 3" "6#/%&y '(0)G\"\"$" }{TEXT -1 2 " ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 298 "s:='s': S:='S': t:='t': Y:='Y':\ny0 := -1:\ny1 := 3:\nF := laplac e(t, t , s):\n`y''(t) + y(t) = t`;\n`y(0) ` = y0, ` y'(0) ` = y1; \neqn := s^2*Y(s) - s*y0 - y1 +1*Y(s) = F:\neqn;\nsol := simplify(sol ve(subs(s=S,eqn),Y(S))):\nY := s -> subs(S=s,sol):\n`Y(s) ` = Y(s);\n` Y(s) ` = convert(Y(s), parfrac, s);" }}}{PARA 11 "" 1 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "Usando que " }{XPPEDIT 18 0 "L(t) = 1/(s^2)" "6#/-%\"LG6#%\"tG*&\"\"\"\"\"\"*$%\"sG\"\"#!\"\"" }{TEXT -1 9 " e " }{XPPEDIT 18 0 "L(2*sin(t)- cos(t)) = (s-2)/(s^2+1)" "6# /-%\"LG6#,&*&\"\"#\"\"\"-%$sinG6#%\"tGF*F*-%$cosG6#F.!\"\"*&,&%\"sGF* \"\"#F2F*,&*$F5\"\"#F*\"\"\"F*F2" }{TEXT 353 1 "\n" }{TEXT -1 31 "a so lu\347\343o \351 combina\347\343o linear ." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 129 "F1 := 1/(s^2):\nF2 := (s-2)/(s^2+1):\n`Y(s) ` = F1 -F2;\nf1 := invlaplace(F1, s , t):\nf2 := invlaplace(F2, s , t):\n`f( t) ` = f1 -f2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 15 "with(inttrans):" }}}{PARA 0 "" 0 "" {TEXT 355 11 "Exemplo 4 " }{TEXT -1 38 " Resolva o problema de valor inici al " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "`y''(t) + y'(t) - y(t) = 4*exp( t)`" "6#%By''(t)~~+~y'(t)~-~y(t)~=~4*exp(t)G" }{TEXT -1 10 " com \+ " }{XPPEDIT 18 0 "y(0) = 1" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 5 " e \+ " }{XPPEDIT 18 0 "`y'(0)` = 0" "6#/%&y'(0)G\"\"!" }{TEXT -1 2 " ." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 331 "s:='s': S:='S': t:='t': Y:= 'Y':\ny0 := 1:\ny1 := 0:\nF := laplace(4*exp(t), t , s);\n`y''(t) + y '(t)-y(t) = 4*exp(t)`;\n`y(0) ` = y0, ` y'(0) ` = y1;\neqn := s^2* Y(s) - s*y0 - y1 +(s*Y(s) - y0)-1*Y(s) = F:\neqn;\nsol := simplify(sol ve(subs(s=S,eqn),Y(S))):\nY := s -> subs(S=s,sol):\n`Y(s) ` = Y(s);\n` Y(s) ` = convert(Y(s), parfrac, s);" }}}{PARA 265 "" 1 "" {TEXT -1 0 " " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 "Algo mais sobre EDO's" }} {PARA 3 "" 0 "" {TEXT -1 35 "M\351todos Num\351ricos para EDO no Maple " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{PARA 0 "" 0 "" {TEXT -1 362 "Se voc\352 est\341 i nteressado no solu\347\343o de um problema de valor inicial em um pont o particular ou em um n\372mero pequeno de pontos particulares, \351 \+ mais eficiente resolver numericamente a equa\347\343o diferencial do q ue tentar resolv\352-la exatamente usando f\363rmulas. Muitas vezes, n \343o existe uma express\343o expl\355cita para a solu\347\343o, e a \355 teremos que usar m\351todos num\351ricos. " }}{PARA 0 "" 0 "" {TEXT -1 70 "Vamos ver agora como se pode obter a solu\347\343o num \351rica usando o Maple." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 211 "O comando dsolve, sem informa\347\343o de op\347 \343o tentar\341 uma solu\347\343o exata. Uusando o comando dsolve co m a op\347\343o \"numeric\" o Maple retorna um procedimento que voc \352 dever\341 usar para obter a solu\347\343o no ponto procurado. " } }{PARA 0 "" 0 "" {TEXT -1 61 "Como default o Maple usa o m\351todo de Runge-Kutta de ordem 4." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 356 11 "Exemplo 1:" }{TEXT -1 23 " Considere a EDO y'= \+ " }{XPPEDIT 18 0 "x^2-y^2" "6#,&*$%\"xG\"\"#\"\"\"*$%\"yG\"\"#!\"\"" }{TEXT -1 16 " , com y(0)=1. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "\{D(y)(x) = x^2 - y^2, y(0)= 1\};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "solucao:=dsolve(\{D (y)(x) = x^2 - y(x)^2, y(0)=1\}, y(x), numeric);" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 102 "O Maple retornou um procedimento que dar\341 a s olu\347\343o no ponto desejado. Para saber a solu\347\343o no ponto " }}{PARA 0 "" 0 "" {TEXT -1 28 "x=1, execute a linha abaixo." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "solucao(1); #d\341 a solu \347\343o no ponto x=1 e etc." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "solucao(2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "soluca o(5);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 307 "N\343o temos nenhum controle, deixando o Maple fazer todas as con tas e tomar todas as decis\365es. Para obter mais controle devemos esp ecificar o m\351todo a ser usado pelo Maple e tamb\351m o passo a ser \+ usado. Existem tr\352s m\351todos num\351ricos que usamos mais comume nte: m\351todo de Euler, M\351todo de Heum e o Runge-Kutta." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Vamos ver como \+ isto pode ser feito." }}{PARA 0 "" 0 "" {TEXT 357 9 "Exemplo 2" } {TEXT -1 34 ": Considere a seguinte EDO y' = " }{XPPEDIT 18 0 " y^2- x" "6#,&*$%\"yG\"\"#\"\"\"%\"xG!\"\"" }{TEXT -1 14 " com y(0)=0. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "edo := D(y)(x) = y(x)^2-x;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "Estimar y(1) se y(0)=0, usando o m\351todo de Euler com passo \+ h = 0.01." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "eulersol:=dsol ve(\{edo, y(0)=0\}, y(x), numeric, method=classical[foreuler], stepsiz e=0.01);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "eulersol(1);" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "Estimar y(1) se y(0)=0, usand o o m\351todo de Euler melhorado com passo h = 0.05." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "Melhoreulersol:=dsolve(\{edo, y(0)= 0\}, y(x), numeric, method=classical[heunform], stepsize=0.05);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Melhoreulersol(1);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "Estimar y(1) se y(0)=0, usando \+ o m\351todo de Runge-Kutta com passo h = 0.1." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "rksol:=dsolve(\{edo, y(0)=0\}, y(x), numeric, method=classical[rk4], stepsize=0.1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "rksol(1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 358 9 "Exem plo 3" }}{PARA 0 "" 0 "" {TEXT -1 98 "Vamos ver agora como usar os rec ursos num\351ricos do Maple para plotar campos de dire\347\365es de ED O's. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "Uma EDO y'(x)= f(x, y) define um campo de dire\347\365es. O Maple po de plotar este campo. Vejamos como se se faz isto. " }}{PARA 0 "" 0 " " {TEXT -1 77 "Antes precisamos carregar o pacote de ferramentas para \+ equa\347\365es diferenciais." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "eqc 1:= diff(y(x),x) + y(x) = 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "ODE:= diff(y(x),x) = sqrt(y(x)^2+1);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 65 "DEplot(eqc1,y(x),x=-4..4,y=-4..4,title=`Exemplo 1`, \narrows=slim);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 " DEplot( ODE,y(x),x=-4..4,y=-4..4,title=`Exemplo 2`,\narrows=slim);" }}}{PARA 0 "" 0 "" {TEXT -1 4 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "eqc2 := diff(y(x),x) = (y(x)-x)*(1-y(x)^3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 " Gerando um campo de dire\347\365es" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "DEplot(eqc2,y(x),x=0..2,y=0.5..2,title=`Exemp lo 3`,arrows=SLIM);" }}}{PARA 0 "" 0 "" {TEXT -1 60 " Podemos tamb \351m plotar a solu\347\343o usando o m\351todo de Euler." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "DEplot(eqc2,y(x),x=0..2,\{[0,2],[0, 0.5]\},\narrows=NONE,method=classical[foreuler]);" }}}{PARA 0 "" 0 "" {TEXT -1 39 " Usando m\351todo de Euler melhorado." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "DEplot(eqc2,y(x),x=0..2,\{[0,2],[0, 0.5]\},\narrows=NONE,method=classical[heunform]);" }}}{PARA 0 "" 0 "" {TEXT -1 42 " Usando Runge-Kutta4, default do Maple." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "DEplot(eqc2,y(x),x=0..2,\{[0,2],[0, 0.5]\},arrows=SMALL);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "DE plot(eqc2,y(x),x=0..2,\{[0,2],[0,0.5]\},\narrows=NONE,method=classical [heunform]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 15 "Alguns Modelos " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(inttrans):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 361 33 "Modelo 1:Desintegra\347\343o Radioativa" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 142 "Se a quantidade do ma terial radiotivo no instante t \351 Q(t),\nent\343o a taxa de varia \347\343o no instante t \351 proporcional a\nquantidade Q(t), isto \+ \351,\n" }{XPPEDIT 18 0 "d/dt *Q(t) = - k* Q(t)" "6#/*(%\"dG\"\"\"%#dt G!\"\"-%\"QG6#%\"tGF&,$*&%\"kGF&-F*6#F,F&F(" }{TEXT -1 72 ",\nonde k>0 \351 chamada a constante de desintegra\347\343o radiotiva do\nmateria l." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "Se a quantidade inicial de material \351 Q[0], ent\343o podemos usar a t ransformada de Laplace para determinarmos a solu\347\343o do PVI:" }} {PARA 0 "" 0 "" {XPPEDIT 18 0 "d/dt*Q(t) = -k*Q(t)" "6#/*(%\"dG\"\"\"% #dtG!\"\"-%\"QG6#%\"tGF&,$*&%\"kGF&-F*6#F,F&F(" }}{PARA 0 "" 0 "" {TEXT -1 6 "Q(0)= " }{XPPEDIT 18 0 "Q[0]" "6#&%\"QG6#\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 247 "s:= 's': S:='S': t:='t': Y:='Y':\ny0 := Q[0]:\nF := 0:\n`y'(t) +ky(t) = \+ 0`;\n`y(0) ` = y0;\neqn := s*Y(s) - y0 +k*Y(s) = F:\neqn;\nsol := sim plify(solve(subs(s=S,eqn),Y(S))):\nY := s -> subs(S=s,sol):\n`Y(s) ` = Y(s);\n`Y(s) ` = convert(Y(s), parfrac, s);" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Usando que " } {XPPEDIT 18 0 "L(exp(-kt)) = 1/(s+k)" "6#/-%\"LG6#-%$expG6#,$%#ktG!\" \"*&\"\"\"\"\"\",&%\"sGF/%\"kGF/F," }{TEXT -1 4 " " }{TEXT 359 1 " \n" }{TEXT -1 31 "a solu\347\343o \351 combina\347\343o linear ." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "F1 := 1/(s+k):\n`Y(s) ` = Q[0]* F1; \nf1 := invlaplace(F1, s , t):\n`f(t) ` = Q[0]*f1 ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Podemos us ar a inversa de Laplace diretamente." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "`Y(s) ` = Y(s);\n`f(t) ` = invlaplace(Y(s), s , t);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 362 36 "Modelo 2: Movime nto num meio viscoso" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 242 "Uma particula de massa m \351 abandonada num meio visc oso, isto \351,\nnum meio que oferece resist\352ncia ao seu movimento \+ proprocional a\nvelocidade. O movimento da part\355cula no meio viscos o depende da\nconstante k de viscosidade, da seguinte forma\nF= " } {XPPEDIT 18 0 "m * a = m*g-k* v" "6#/*&%\"mG\"\"\"%\"aGF&,&*&F%F&%\"gG F&F&*&%\"kGF&%\"vGF&!\"\"" }{TEXT -1 58 ",\nonde a\351 a acelera\347 \343o e v \351 a velocidade.\n\nReescrevendo,\n" }{XPPEDIT 18 0 "(dv)/ (dt)+k*m*v=g" "6#/,&*&%#dvG\"\"\"%#dtG!\"\"F'*(%\"kGF'%\"mGF'%\"vGF'F' %\"gG" }{TEXT -1 15 ".\n\nSupondo que " }{XPPEDIT 18 0 "v(0)=v[0]" "6# /-%\"vG6#\"\"!&F%6#F'" }{TEXT -1 40 ", obtemos a solu\347\343o via Lap lace dada por" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(inttrans):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 252 "s:='s': S:='S': t:='t': Y:='Y':\ny 0 := v[0]:\nF := g/s:\n`y'(t) +kmy(t) = g`;\n`y(0) ` = y0;\neqn := \+ s*Y(s) - y0 +k*m*Y(s) = F:\neqn;\nsol := simplify(solve(subs(s=S,eqn), Y(S))):\nY := s -> subs(S=s,sol):\n`Y(s) ` = Y(s);\n`Y(s) ` = convert( Y(s), parfrac, s);" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 13 "Usando que " }{XPPEDIT 18 0 "L(exp(-kmt)) = 1/(s+km) " "6#/-%\"LG6#-%$expG6#,$%$kmtG!\"\"*&\"\"\"\"\"\",&%\"sGF/%#kmGF/F," }{TEXT -1 5 " e " }{XPPEDIT 18 0 "L(1) = 1/(s)" "6#/-%\"LG6#\"\"\"*& \"\"\"\"\"\"%\"sG!\"\"" }{TEXT -1 1 " " }{TEXT 360 1 "\n" }{TEXT -1 31 "a solu\347\343o \351 combina\347\343o linear ." }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 180 "F1 := 1/(s+k*m):\nF2 := 1/(s):\n`Y(s) ` = ( k*m*v[0]-g)/(k*m)* F1+ (g/k*m)*F2;\nf1 := invlaplace(F1, s , t):\nf2 : = invlaplace(F2, s , t):\n`f(t) ` = (k*m*v[0]-g)/(k*m)*f1 + (g/k*m)*f2 ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Podemos usar a inversa de L aplace diretamente." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "`Y(s) ` = Y( s);\n`f(t) ` = invlaplace(Y(s), s , t);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 363 40 "Modelo 3: Lei de resfriamenteo de Newton" }{TEXT -1 76 "\n\nA lei de resfriamente de Newton diz que a t axa de varia\347\343o da\ntemperatura " }{XPPEDIT 18 0 "(dT)/(dt)" "6# *&%#dTG\"\"\"%#dtG!\"\"" }{TEXT -1 107 " de um corpo em rela\347\343o \+ ao\ntempo \351 proporcional a diferen\347a da sua temperatura T e da\n temperatura ambiente " }{XPPEDIT 18 0 "T[0]" "6#&%\"TG6#\"\"!" }{TEXT -1 11 " , isto \351,\n" }{XPPEDIT 18 0 "(dT)/(dt)= -k*(T-T[0])" "6#/*& %#dTG\"\"\"%#dtG!\"\",$*&%\"kGF&,&%\"TGF&&F-6#\"\"!F(F&F(" }{TEXT -1 55 ",\nonde k>0 \351 uma constante que depende do material.\n\n. " }} {PARA 0 "" 0 "" {TEXT -1 12 "Supondo que " }{XPPEDIT 18 0 "T(0)= C" "6 #/-%\"TG6#\"\"!%\"CG" }{TEXT -1 40 ", obtemos a solu\347\343o via Lapl ace dada por" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(inttrans):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 255 "s:='s': S:='S': t:='t': Y:='Y':\ny 0 := C:\nF := k*T[0]/s:\n`y'(t) +ky(t) = k*T[0]`;\n`y(0) ` = y0;\neqn := s*Y(s) - y0 +k*Y(s) = F:\neqn;\nsol := simplify(solve(subs(s=S,eq n),Y(S))):\nY := s -> subs(S=s,sol):\n`Y(s) ` = Y(s);\n`Y(s) ` = conve rt(Y(s), parfrac, s);" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Usando que " }{XPPEDIT 18 0 "L(exp(-k*t))= 1/(s+k) " "6#/-%\"LG6#-%$expG6#,$*&%\"kG\"\"\"%\"tGF-!\"\"*&\"\"\"F-,&%\"sGF-F ,F-F/" }{TEXT -1 13 " e " }{XPPEDIT 18 0 "L(1) = 1/(s)" "6#/ -%\"LG6#\"\"\"*&\"\"\"\"\"\"%\"sG!\"\"" }{TEXT -1 1 " " }{TEXT 364 3 " \n " }{TEXT -1 14 "a solu\347\343o \351 ." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 149 "F1 := 1/(s):\nF2 := 1/(s+k):`Y(s) ` = T[0]*F1+(-T[ 0]+C)*F2;\nf1 := invlaplace(F1, s , t):\nf2 := invlaplace(F2, s , t): \n`f(t) ` = T[0]*f1+(-T[0]+C)*f2 ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Podemos usar a inversa de Laplace diretamente." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "`Y(s) ` = Y(s);\n`f(t) ` = invlaplace(Y(s), s , \+ t);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 365 18 "Modelo 4 : P\352ndulo" }}{PARA 266 "" 0 "" {OLE 1 252948 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:JZ:F =F=N=V=^=f=n=v=>>AF ANAVA^AfAnA;jYJZB:=EFE NEVE^EfEnEvE>F:::::::JEf:yyyxIN::;`:Z@[::Jj]Mtd x:Ym[Aj;J:@:<:=j[vGUMrvC?MoJ::::::::JCNZ;^:vYxI>:<::::::j`J:j:vCSmlF@[ KaFFcmnnHEM:>:::::::oJ;@j:j;B:yayA:<::::::IZ:^vBY:>:=:jRj^^HEmpnCfGEM: >::::::n=?R:yyyyyy:>:<::::::?:EJ:>:F:;JyKy;vY::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::F:wyyAbR<:Tn=JbDNsPqs\\`A:::ZnIalHH>::::::::::j:J:::::::::::=Zk>K?C:Q 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:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::B:Zy=::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::<:ry::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::Z::xI:::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::B:>:::::::::::::::::::::::::::::::::::::::::::::::::j:vC:;:::: :::::N[:NZ:vyyuy:>:<::::::AB:;JD<:;::::::::::::vYxI:;Z:::::::::::::::: ::::yay=J:B:::::::::::::::::::jysy:>:<:::::::::::3:" }}{PARA 0 "" 0 " " {TEXT -1 479 "O diagrama mostra um p\352endulo consistindo de uma pe quena massa\nesf\351rica suspensa por um fio. O p\352ndulo est\341 ime rso em um\nl\355quido que resiste ao seu movimento. A flecha verde mos tra a\nvelocidade v da esfera, a flecha vermelha mostra a for\347a\na gindo sobre ela.\n\nExiste uma for\347a para baixo g devido a gravida de e tamb\351m\nr uma for\347a resistiva devido ao fluido, agindo na d ire\347\343o oposta a velocidade do p\352ndulo.\n\n\nA equa\347\343o \+ que descreve o movimento do p\352ndulo em termos do\n\342ngulo " } {XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 54 " entre a esfera do p \352ndulo e a vertical \351 dada por:\n" }{XPPEDIT 18 0 "theta [tt] = - k * theta[t] -theta (t)" "6#/&%&thetaG6#%#ttG,&*&%\"kG\"\"\"&F%6# %\"tGF+!\"\"-F%6#F.F/" }{TEXT -1 184 ".\nA constante k \351 a medid a que mostra quanta resist\352ncia o\nfluido exerce sobre a esfera. Se k = 0 ent\343o o fluido n\343o \nexerce resist\352ncia.\nTomando as \+ condi\347\365es iniciais dadas por " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "theta(0)= phi" "6#/-%&thetaG6#\"\"!%$phiG" }{TEXT -1 7 ", e " } {XPPEDIT 18 0 "theta [t](0)=0" "6#/-&%&thetaG6#%\"tG6#\"\"!F*" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 58 "obtemos via transformada de \+ Laplace aa seguinte solu\347\343o.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(inttrans):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 296 "s:='s' : S:='S': t:='t': Y:='Y':\ny0 := phi:\ny1 := 0:F := 0:\n`y''(t) +ky'( t)+y(t) = 0`;\n`y(0) ` = y0, ` y'(0) ` = y1;\neqn := s^2*Y(s) -s*y0 -y1 +k*(s*Y(s)-y0)+Y(s) = F:\neqn;\nsol := simplify(solve(subs(s=S,eqn ),Y(S))):\nY := s -> subs(S=s,sol):\n`Y(s) ` = Y(s);\n`Y(s) ` = conver t(Y(s), parfrac, s);" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Podemos usar a inversa de Laplace diretam ente." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "`Y(s) ` = Y(s);\n`f(t) ` = simplify(invlaplace(Y(s), s , t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "10" 0 }{VIEWOPTS 1 1 0 1 1 1803 }