Question

Find the inverse Laplace transform for \(\frac{s}{(s+2)^2}\).

(a) te^-t u(t)

(b) e^-t  sin⁡t u(t)

(c) e^-2t (1-2t)u(t)

(d) e^2t (1-2t)u(t)

I have been asked this question in a job interview.

This intriguing question comes from Inverse Laplace Transform topic in portion Laplace Transform and System Design of Signals and Systems

Answer 1

Right option is (c) e^-2t (1-2t)u(t)

Easy explanation: Given X(s) = \(\frac{s}{(s+2)^2}\)

x(t) = L^-1 [X(s)] = \(L^{-1} \Big[\frac{s}{(s+2)^2}\Big] = L^{-1} \Big[\frac{s+2}{(s+2)^2} – \frac{2}{(s+2)^2}\Big] = L^{-1} \Big[\frac{1}{s+2}\Big] – 2L^{-1} \Big[\frac{1}{(s+2)^2}\Big] \)

= e^-2t – 2e^-2t L^-1 \([\frac{1}{s^2}]\) = e^-2t (1-2t)u(t).