Inverse Laplace transform of $R(s)$

Question

Calculate the Inverse Laplace transform of $$R(s)=\frac{s^3+s^2+2}{s^2-1}.$$

I have split it into $$\frac{s^3+s^2+2}{s^2-1}=s+1-\frac{1}{s+1}+\frac{2}{s-1},$$ but I am facing difficulty to find the Inverse Laplace Transform of the first part $$s+1.$$

Thanks in advance.

Answer 1

I agree with your partial fraction expansion. We have:

$$s+1 + \dfrac{2}{s-1}-\dfrac{1}{s+1}$$

Using this table of Laplace transforms, we find:

$$\mathscr{L}^{-1}\left(s+1 + \dfrac{2}{s-1}-\dfrac{1}{s+1}\right) = \delta'(t) + \delta(t) + 2 e^t - e^{-t}$$

You are going to have to think through the first two as they are a bit tricky.