I have been trying to teach myself about Laplace Transformations before I take Differential Equations in the spring and I ran into a problem that I'm not sure how to solve.
If an undamped system is given a force that consists of two piecewise functions, like $f(t) = u(t) - u(t-1)$, do you take the laplace transform of each function separately, i.e. $\mathscr{L}[f(t)] = \mathscr{L}[u(t)] - \mathscr{L}[u(t-1)]$?
Or do you need to manipulate $u(t)$ so it has the same $a$ as $u(t-1)$ before you find the transform, assuming $u(t-1)$ is in the form $u(t-a)$.
If it is the first case, $\mathscr{L}[u(t)] = \frac{1}{s}$, correct?
I'm aware that this could potentially be a trivial issue but I just want to confirm that I have the right approach before I move on.
Thanks!
You can take the Laplace transform separately, since the Laplace transform is a linear operator. Which means,
$$\mathcal L [\alpha f(x)+\beta g(x)]=\alpha\mathcal L [ f(x)]+\beta\mathcal L[ g(x)]$$ for $\alpha, \beta \in \mathbb R$