The question is to solve the following initial conditions problem with the Laplace transform method.
$$ f'' + 2f' -3f = \begin{cases} 1, \ 0 \leq t < c \\ 0, \ t \geq c \end{cases}; f(0) = f'(0) = 0 $$
What I did was apply the Laplace transform to both sides so we get:
$$ \mathcal{L}\{ f \}(s) \cdot (s^2 + 2s - 3) = \int_0^c e^{-st} dt = \frac{1-e^{-sc}}{s} \implies \mathcal{L}\{ f \} = \frac{1 - e^{-sc}}{s(s+3)(s-1)} $$
I have no idea how to continue from here. I wasn't able to figure out the inverse transform from the basic properties (Linearity, derivative, primitive, frequency translation, time translation, etc).
Thanks in advance for any responses.