Equações Diferenciais Transformada de Laplace€¦ · Equações Diferenciais Transformada de...

Equações Diferenciais

Transformada de Laplace

Tiago [email protected]

ICMC

mailto:[email protected]?subject=

O que é a transformada de Laplace

https://sites.icmc.usp.br/tiago/index.html/edo.html

Ferramenta para resolução de EDO




1785 estudando Teoria de Probabilidade




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Tempo




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Tempo “s”




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+ +


Alguns exemplos


L+ +

Operação Linear




Brief Article

The Author

April 15, 2020

sinal Transformada

f(t) F (s)

eat1

s� a

11

s

f 0(t) sF (s)� f(0)

Table 1: Remember, never use vertical lines in tables.

1


Resolvendo uma EDO por Laplace


x

0 + x = 0

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Recap

x(t) = x0e�t

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x

0 + x = 0

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L(x0 + x) = L(0)

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L(x0) + L(x) = 0


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Por Laplace




Por Laplace

Brief Article

The Author

April 15, 2020

sinal Transformada

f(t) F (s)

eat1

s� a

11

s

f 0(t) sF (s)� f(0)


1

L(x0) + L(x) = 0


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aFSMKeAyMTNSBBkaPf3b6gc48ogvMUNCdE0jlH

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sX(s)� x0 +X(s) = 0


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lsrN67yOIpwCEdQARvq0IAbaEILCEzgGV7hzXgyX

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sX(s)� x0 +X(s) = 0

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X(s)(1 + s) = x0


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x

0 + x = 0

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Por Laplace




x

0 + x = 0

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Por Laplace

X(s) = x01

s+ 1


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Por Laplace

X(s) = x01

s+ 1


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Brief Article

The Author

April 15, 2020

sinal Transformada

f(t) F (s)

eat1

s� a

11

s

f 0(t) sF (s)� f(0)


1

) x0e�t


Recap


EDOx(t)=? L

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atexit>

Formula X(s)


Recap


Achax(t) L

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atexit>

Formula X(s)


Pergunta


L

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atexit>

O que tem dentro dessa caixa?

Como funciona essa transformação


Detour


Calcular a area da região

0 1

e-sx


Detour


Calcular a area da região

A area depende de s

Area = A(s)

0 1

e-sx



Como integramos em x a resposta não depende de x

A(s) =

Z 1

0e�sx

Detour



A(s) =

Z 1

0e�sx

A(s) = �1se�sx

��1

0

A(s) =1

s

Detour




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atexit>

sinalf(t)




sinalf(t)

Area sob

e�stf(t)




sinalf(t)

Area sob Formula F(s)e�stf(t)



sinal1

Area sob

e�stf(t)



sinal1

Area sob Formula e�stf(t) 1

s



sinal1

Area sob Formula e�stf 1

s

Brief Article

The Author

April 15, 2020

sinal Transformada

f(t) F (s)

eat1

s� a

11

s

f 0(t) sF (s)� f(0)


1

Exemplo


Resolva PVIx

0 + 5x = e�2t

x(0) = 0

Exemplo


x

0 + 5x = e�2t

sX(s)� x0 + 5X(s) =1

s+ 2

L(x0 + 5x) = L(e�2t)

AAACHXicbVDJSgNBEO2JW4zbqEcvjUFIEMNMiMSLEPCgBw8RzALZ6OlUkiY9C909kjDMj3jxV7x4UMSDF/Fv7CyHGH1Q8Hiviqp6TsCZVJb1bSRWVtfWN5Kbqa3tnd09c/+gKv1QUKhQn/ui7hAJnHlQUUxxqAcCiOtwqDnDq4lfewAhme/dq3EALZf0PdZjlCgtdcxC0yVqQAmPbuPMqB01A8FciPEpPh/hLL7Eiz60o7O8irMdM23lrCnwX2LPSRrNUe6Yn82uT0MXPEU5kbJhW4FqRUQoRjnEqWYoISB0SPrQ0NQjLshWNP0uxida6eKeL3R5Ck/VxYmIuFKOXUd3Tm6Vy95E/M9rhKp30YqYF4QKPDpb1As5Vj6eRIW7TABVfKwJoYLpWzEdEEGo0oGmdAj28st/STWfswu54l0hXbqex5FER+gYZZCNiqiEblAZVRBFj+gZvaI348l4Md6Nj1lrwpjPHKJfML5+AG33oOI=

Exemplo


x

0 + 5x = e�2t

sX(s)� x0 + 5X(s) =1

s+ 2

(s+ 5)X(s) =1

s+ 2

L(x0 + 5x) = L(e�2t)

AAACHXicbVDJSgNBEO2JW4zbqEcvjUFIEMNMiMSLEPCgBw8RzALZ6OlUkiY9C909kjDMj3jxV7x4UMSDF/Fv7CyHGH1Q8Hiviqp6TsCZVJb1bSRWVtfWN5Kbqa3tnd09c/+gKv1QUKhQn/ui7hAJnHlQUUxxqAcCiOtwqDnDq4lfewAhme/dq3EALZf0PdZjlCgtdcxC0yVqQAmPbuPMqB01A8FciPEpPh/hLL7Eiz60o7O8irMdM23lrCnwX2LPSRrNUe6Yn82uT0MXPEU5kbJhW4FqRUQoRjnEqWYoISB0SPrQ0NQjLshWNP0uxida6eKeL3R5Ck/VxYmIuFKOXUd3Tm6Vy95E/M9rhKp30YqYF4QKPDpb1As5Vj6eRIW7TABVfKwJoYLpWzEdEEGo0oGmdAj28st/STWfswu54l0hXbqex5FER+gYZZCNiqiEblAZVRBFj+gZvaI348l4Md6Nj1lrwpjPHKJfML5+AG33oOI=

Exemplo


X(s) =1

s+ 5

1

s+ 2

Exemplo


X(s) =1

s+ 5

1

s+ 2

Massagear X(s) até ele virar algo conhecido

Brief Article

The Author

April 15, 2020

sinal Transformada

f(t) F (s)

eat1

s� a

11

s

f 0(t) sF (s)� f(0)


1

Exemplo: Frações Parciais


X(s) =1

s+ 5

1

s+ 2

=A

s+ 5+

B

s+ 2

A+B = 0

2A+ 5B = 1

A = �1/3 B = 1/3

A = �1/3 B = 1/3


X(s) =1

3

✓1

s+ 2� 1

s+ 5

◆

x(t) =1

3

�e

�2t � e�5t�

Solução do PVI

Exemplo: Frações Parciais

Equações Diferenciais Transformada de Laplace€¦ · Equações Diferenciais Transformada de...

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