How to find the Laplace inversion of (p)/(p^4+4)?

inurbandojoa

inurbandojoa

Answered question

2022-11-18

How to find the Laplace inversion of p p 4 + 4 ?

Answer & Explanation

motylowceyvy

motylowceyvy

Beginner2022-11-19Added 19 answers

p 4 + 4 has 4 roots: p = ± 1 ± i, so it can be written as: ( p 1 i ) ( p 1 + i ) ( p + 1 i ) ( p + 1 + i ) = ( p 2 2 p + 2 ) ( p + 2 p + 2 ), so:
p p 4 + 4 = p ( p 2 2 p + 2 ) ( p + 2 p + 2 ) = A p + B p 2 2 p + 2 + C p + D p 2 + 2 p + 2
The solution is A = 0 , B = 1 4 , C = 0 , D = 1 4 (I can show more detail of the computation if you need it), so:
p p 4 + 4 = 1 4 p 2 2 p + 2 1 4 p 2 + 2 p + 2 = 1 4 ( p 1 ) 2 + 1 1 4 ( p + 1 ) 2 + 1
We know that L { e a x f ( x ) } = F ( p a ) , L { sin ( a x ) } = a p 2 + a 2 , L { cos ( a x ) } = p p 2 + a 2 , so:
L 1 { 1 4 ( p 1 ) 2 + 1 } = 1 4 e x sin x
L 1 { 1 4 ( p + 1 ) 2 + 1 } = 1 4 e x sin x
Finaly:
L 1 { p p 4 + 4 } = 1 4 e x sin x 1 4 e x sin x
L 1 { p p 4 + 4 } = 1 4 sin x ( e x e x )
L 1 { p p 4 + 4 } = 1 4 sin x ( 2 sinh x )
L 1 { p p 4 + 4 } = 1 2 sin x sinh x

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?