How do I find Laplace transform of 7t * e^(-3t) * sin(3t)?

klastiesym

klastiesym

Answered question

2022-10-22

How do I find Laplace transform of 7 t e 3 t sin ( 3 t )?

Answer & Explanation

dwubiegrw

dwubiegrw

Beginner2022-10-23Added 13 answers

Here is one approach (there are others)
L     ( sin 3 t ) = 3 s 2 + 3 2 = 3 s 2 + 9
L     ( e 3 t sin 3 t ) = 3 ( ( s + 3 ) 2 + 9 ) , via frequency shifting.
So, L     ( 7 e 3 t sin 3 t ) = 21 ( s + 3 ) 2 + 9
Finally, we use frequency differentiation.
L     ( t   7 e 3 t sin 3 t ) = d d s [ 21 ( s + 3 ) 2 + 9 ] = 42 ( s + 3 ) ( ( s + 3 ) 2 + 9 ) 2
Tara Mayer

Tara Mayer

Beginner2022-10-24Added 4 answers

Hints: Use formal rules L ( t f ( t ) ) = F ( s ), and combine it with the following table result:
L ( e λ t sin ( μ t ) ) = 0 e ( s + λ ) t e i μ t d t = ( 1 s + λ i μ ) = μ ( s + λ ) 2 + μ 2

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