Inverse Laplace Transform of s^(-2)(s^2+1)^(-1) Using Convolution Theorem?

zaiskaladu

zaiskaladu

Answered question

2022-09-25

Inverse Laplace Transform of s 2 ( s 2 + 1 ) 1 Using Convolution Theorem?

Answer & Explanation

Abagail Meyers

Abagail Meyers

Beginner2022-09-26Added 10 answers

Let F ( s ) = s 2 ( s 2 + 1 ) 1 and let L ( f ( t ) ) = F ( s ) We know that
L ( t ) = s 2
and
L ( sin t ) = ( s 2 + 1 ) 1
So
F ( s ) = L ( f ( t ) ) = L ( t sin t ) = L ( t ) L ( sin t ) = s 2 ( s 2 + 1 ) 1
where ∗ stands for convolution. So, the time domain laplace inverse is really
f ( t ) = t sin t = 0 t τ sin ( t τ )   d τ = 0 t ( t τ ) sin ( τ )   d τ = t sin ( t )

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