What is the Laplace Inverse of the given question? (sqrt(4+s^3))/(s^3)

independanteng

independanteng

Answered question

2022-10-18

What is the Laplace Inverse of the given question?
4 + s 3 s 3
I tried solving it by expanding 1 + s 3 4 but the terms will not have Laplace Inverse. If I expand it like 1 + 4 s 3 i.e.
s 3 / 2 1 + 4 s 3
It will still have many terms of t n . Are there other methods?

Answer & Explanation

getrdone07tl

getrdone07tl

Beginner2022-10-19Added 23 answers

With computer algebra system help I have:
L s 1 [ 4 + s 3 s 3 ] ( t ) = 2 t 1 F 3 ( 1 2 ; 1 2 , 5 6 , 7 6 ; 4 t 3 27 ) π
For general a:
L s 1 [ a + s 3 s 3 ] ( t ) = L s 1 [ n = 0 a n ( s 3 ) 1 2 n ( 1 2 n ) ] ( t ) = n = 0 L s 1 [ a n ( s 3 ) 1 2 n ( 1 2 n ) ] ( t ) = n = 0 a n π t 1 2 + 3 n 2 Γ ( 3 2 n ) Γ ( 1 + n ) Γ ( 3 2 + 3 n ) = 2 t 1 F 3 ( 1 2 ; 1 2 , 5 6 , 7 6 ; 1 27 ( a t 3 ) ) π

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