What is the inverse laplace transform of $\large{\frac{ s^3 - a^2s }{(s^2 + a^2)^2}}$

Question

I tried convolution and partial fractions but both turned out to be too much work. Is there any easy work around??

Answer 1

Partial fraction expansion

$$\frac{s}{a^2+s^2}-\frac{2 a^2 s}{\left(a^2+s^2\right)^2}$$

Notice that

$$\frac{d}{ds}\frac{a^2}{a^2+s^2}=-\frac{2 a^2 s}{\left(a^2+s^2\right)^2}$$

Use the transform table and the derivative rule to find the result as

$$\cos (a t)-a t \sin (a t)$$

This is a somewhat difficult problem I think