What is the Laplace transform of e^(3t) * sin^2 t

Antwan Perez 2022-10-23 Answered
What is the Laplace transform of e 3 t sin 2 t
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Answers (1)

Laci Conrad
Answered 2022-10-24 Author has 17 answers
Converted my comments into an answer:
The procedure I would use is to find the first find the Laplace Transform of sin 2 ( t ) using the identity sin 2 ( t ) 1 cos ( 2 t ) 2 and then apply the first shifting theorem, which states that:
(1) L { e a t f ( t ) } = F ( s a )
Where F ( s ) = L { f ( t ) }. It is easy to prove the above using the definition of the Laplace Transform, which I leave as an exercise.
Letting f ( t ) = sin 2 ( t ), we obtain for F(s):
F ( s ) = L { sin 2 ( t ) } = 1 2 L { 1 } 1 2 L { cos ( 2 t ) } = 1 2 s s 2 ( s 2 + 4 ) = 2 s ( s 2 + 4 )
Thus, it follows from the first shifting theorem that:
L { e 3 t f ( t ) } = F ( s 3 ) = 2 ( s 3 ) ( ( s 3 ) 2 + 4 ) = 2 ( s 3 ) ( s 2 6 s + 13 )
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