hey, I'm looking at an exercise about solving a differential equation using the Laplace transform. I get the (correct) result, that
$$\mathcal{L}[y] = \frac{1}{(1+s^2)^2} e^{-\pi s} + \frac{1}{(1+s^2)^2}.$$
now, the standard solution suggests to "know" the value of
$$\mathcal{L}^{-1} \left[\frac{1}{(1+s^2)^2}\right]$$
which I don't. Instead, I rewrote my result as
$$\mathcal{L}[\sin(t)] \cdot \mathcal{L}[\sin(t-\pi)] + \mathcal{L}[\sin(t)]^2,$$
which is
$$\mathcal{L}[\sin(t)] (\mathcal{L}[\sin(t-\pi)] + \mathcal{L}[\sin(t)]).$$
