How to calculate Laplace inverse of 1(s2+1)2?

batejavizb

batejavizb

Answered question

2022-09-21

How to calculate Laplace inverse of 1 ( s 2 + 1 ) 2 ?

Answer & Explanation

Julien Zuniga

Julien Zuniga

Beginner2022-09-22Added 7 answers

F ( s ) = 1 ( s 2 + 1 ) 2 = 2 s ( s 2 + 1 ) 2 × 1 2 s
F ( s ) = d d s ( 1 ( s 2 + 1 ) ) × 1 2 s
F ( s ) = G ( s ) H ( s )
And their inverse Laplace Transform are:
g ( t ) = t sin t , h ( t ) = 1 2
Then use the convolution theorem. Evaluate this integral:
f ( t ) = 0 t g ( τ ) h ( t τ ) d τ
f ( t ) = 0 t τ sin ( τ ) × 1 2 d τ
For the Convolution Theorem look a formula ( 0 t f ( t τ ) g ( τ ) d τ = F ( s ) G ( s ))
You should find:
f ( t ) = 1 2 ( sin t t cos t )

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