Inverse Laplace Using Heaviside Function

Question

I have a function for which I need to both find the inverse Laplace transformation and sketch a graph. The function is $$ F(s)=\frac{2}{s^3}-\frac{4}{s^2}e^{-s}-\frac{2}{s^3}e^{-2s} $$ I've gotten as far as finding $$ f(t)=t^2+(-4t+4)(u_1(t)-u_2(t))-t^2u_2(t) $$ using the Heaviside function. I'm not sure how to convert this to a piecewise function. Any help is greatly appreciated!

Answer 1

Inverse Laplace Transform gives: $$f(t)=t^2-(4t-4)\Theta(t-1)-(t-2)^2\Theta(t-2),$$ where $\Theta$ is the Heaviside function. Now you should consider three cases: $t\le 1$, $1<t\le 2$ and $t>2$. Looking at different parts of the function:

$$f(t) = \begin{cases} t^2 : & 0\le t \le 1\\ t^2-4t+4 : & 1 < t\le 2\\ 0 : & t>2. \end{cases}$$ Since this is an Inverse Laplace Transform, we have to set $f$ for all $t\in\mathbb{Z}^-$ to $0$.