Find inverse Laplace Transform of : (1)/((s^2+a^2)^2)

cochetezgh 2022-09-12 Answered
Find inverse Laplace Transform of : 1 ( s 2 + a 2 ) 2
You can still ask an expert for help

Want to know more about Laplace transform?

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (2)

gasskadeu7
Answered 2022-09-13 Author has 21 answers
As
F ( s ) = 0 e s t f ( t ) d t
we have
d d s F ( s ) = d d s 0 e s t f ( t ) d t = 0 e s t t f ( t ) d t
then knowing that L 1 ( 1 s 2 + a 2 ) = 1 a sin ( a t )
d d s ( 1 s 2 + a 2 ) = 0 e s t t a sin ( a t ) d t
then
2 s ( s 2 + a 2 ) 2 t a sin ( a t )
and
1 2 s 2 s ( s 2 + a 2 ) 2 1 2 0 t τ a sin ( a τ ) d τ = sin ( a t ) a t cos ( a t ) 2 a 3

We have step-by-step solutions for your answer!

tophergopher3wo
Answered 2022-09-14 Author has 5 answers
1 ( s 2 + a 2 ) 2 = 1 ( s + i a ) 2 ( s i a ) 2 = 1 4 a 2 [ 1 s + i a 1 s i a ] 2 = 1 4 a 2 [ 1 ( s + i a ) 2 2 1 ( s + i a ) ( s i a ) + 1 ( s i a ) 2 ] = 1 4 a 2 [ 1 ( s + i a ) 2 1 i a ( 1 s i a 1 s + i a ) + 1 ( s i a ) 2 ] = 1 4 a 2 d d s [ 1 s + i a + 1 s i a ] + 1 4 i a 3 ( 1 s i a 1 s + i a ) = 1 4 a 2 d d s 0 e ( s + i a ) t + e ( s i a ) t d t + 1 4 i a 3 0 e ( s i a ) t e ( s + i a ) t d t = 1 4 a 2 0 t ( e ( s + i a ) t + e ( s i a ) t ) d t + 1 4 i a 3 0 e ( s i a ) t e ( s + i a ) t d t = 0 e s t [ 1 4 a 2 ( t e i a t t e i a t ) + 1 4 i a 3 ( e i a t e i a t ) ] d t = 0 e s t [ 1 2 a 2 t cos ( a t ) + 1 2 a 3 sin ( a t ) ] d t
So the inverse Laplace transform is
1 2 a 2 t cos ( a t ) + 1 2 a 3 sin ( a t ) .

We have step-by-step solutions for your answer!

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

New questions