Inverse Laplace Transform Using Piecewise Continuous Functions

Question

Compute the inverse transform of $\displaystyle F(s) = \frac{e^{-2s}}{s^2}$ using unit step functions. Write your answer as a piecewise continuous function.

I don't understand how to do this with piecewise functions.

Answer 1

We have

$$G(s) = \frac{e^{-2s}}{s^2}$$

We know that

$$\mathcal{L}^{-1}\left ( \frac{1}{s^2}\right) = tu(t)$$

and that

$$\mathcal{L}^{-1}\left ( e^{-as}F(s)\right) = f(t-a)u(t-a) \tag{time shift property}$$

Therefore,

$$\mathcal{L}^{-1}\left ( G(s)\right) = (t-2)u(t-2) = \begin{cases} 0 & t \leq 2 \\ t-2 & t > 2 \end{cases}$$