Identifying Reverse Laplace Transforms: A Step-by-Step Guide

[BOAS]

Homework Statement

Hello,

I have just started studying Laplace transformations and I am struggling to identify reverse Laplace transforms. I understand how to perform the transform, but going the other way is really confusing me.

i.e, given $F(P)$ find $f(t)$.

If I have that $F(P) = \frac{5 - 2P}{P^{2} + P - 2}$, and have been told to use the facts that;

$\frac{e^{-at} - e^{-bt}}{b - a} = \frac{1}{(P+a)(P + b)}$ and

$\frac{ae^{-at} - be^{-bt}}{a - b} = \frac{P}{(P+a)(P + b)}$

to find f(t)

Homework Equations

The Attempt at a Solution

I think a good first step is to recognise that the denominator can be factorised;

$F(P) = \frac{5 - 2P}{(P - 1)(P + 2)}$

but I am very confused about what I need to be looking for to move forward. I would really appreciate some guidance here.

Thank you

 

[Orodruin]

Split it in two terms, one with the numerator proportional to P and one constant, and just identify with the terms in the hint you got.