How do I solve the following inverse Laplace transform?

Question

I was solving a physics problem and eventually came upon this transform and I could not find a way to resolve it. $c$ is just a constant. Could anyone help me on this? I appreciate the help. $$ \mathcal{L}_s^{-1}\left[\frac{1}{\left(s^2+1\right)\left(s^2-cs\right)}\right]. $$

Answer 1

First let's think which values can take c. If $c=0$ then easy $$F(s)=\frac{1}{s^2(s^2+1)} = \frac{A}{s}+\frac{B}{s^2}+\frac{Cs+D}{s^2+1}$$ for some constants and the inverse laplace transform should look like $$\mathscr{L}_s^{-1}\{F(s)\}=A+Bt+C\cos t+D\sin t$$ Now if $c\neq0$, easy too $$F(s)=\frac{1}{(s^2+1)s(s-c)}=\frac{A}{s}+\frac{B}{s-c}+\frac{Cs+D}{s^2+1}$$ and the inverse laplace transform should be $$\mathscr{L}_{s}^{-1}\{F(s)\}=A+Be^{ct}+C\cos t+D\sin t$$