How to compute the inverse Laplace transformation of (s^2)/((s^2+1)^2)

metal1fc

metal1fc

Answered question

2022-09-12

How to compute the inverse Laplace transformation of s 2 ( s 2 + 1 ) 2

Answer & Explanation

Illuddybopylu

Illuddybopylu

Beginner2022-09-13Added 17 answers

F ( s ) = s 2 ( s 2 + 1 ) 2 = s 2 + 1 ( s 2 + 1 ) 2 1 ( s 2 + 1 ) 2
F ( s ) = 1 s 2 + 1 + 1 2 s d d s 1 ( s 2 + 1 ) .
Apply inverse Laplace Transform:
f ( t ) = sin t 1 2 0 t 1 × τ sin τ   d τ
f ( t ) = 1 2 ( sin t + t cos t )
Skye Vazquez

Skye Vazquez

Beginner2022-09-14Added 4 answers

Consider the Laplace transforms of sin ( a t ) and t cos ( a t ) which are
sin ( a t ) a s 2 + a 2 = a s 2 + a 3 ( s 2 + a 2 ) 2 t cos ( a t ) s 2 a 2 ( s 2 + a 2 ) 2 .
Now one can notice that
f ¯ ( s ) g ¯ ( s ) 0 t f ( t u ) g ( u ) d u
A second method is to use the convolution theorem,
f¯(s)g¯(s)≑∫t0f(t−u)g(u)du
where f ¯ ( s ) is the transform of f ( t ). Since
cos ( a t ) s s 2 + a 2
then
s 2 ( s 2 + a 2 ) 2 0 t cos ( a u ) cos ( a t a u ) d u [ 2 a t cos ( a t ) sin ( a ( t 2 u ) ) 4 a ] 0 t 1 2 a ( sin ( a t ) + a t cos ( a t ) ) .
Modified form s 3 ( s 2 + 1 ) 2
By using :
cos ( a t ) a t sin ( a t ) s ( s 2 a 2 ) ( s 2 + a 2 ) 2 cos ( a t ) + a t sin ( a t ) s ( s 2 + 3 a 2 ) ( s 2 + a 2 ) 2
then it can be shown that
1 2 ( 2 cos ( a t ) a t sin ( a t ) ) s 3 ( s 2 + a 2 ) 2 .

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Differential Equations

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?