I have a signal which is a sum of $N$ cosines that oscillate between 0 and 1 and that start at different times. More specifically, I have
$$N(t) = \sum_{n=1}^N H(t - t_n) \Big[\frac{1}{2} + \frac{1}{2} \cos(\omega(t - t_n)) \Big]$$
where $H()$ is the Heaviside step function, $\{t_n\}_{n=1}^N$ are known, fixed scalars all greater than 0 and $N(0) = 0$. I'd like to know what second order linear differential equation has this function $N(t)$ as its solution. How could I figure this out?
Disclaimer: Despite how my question may sound, I promise it isn't a homework problem. It's a problem that I want to solve for my research in Bayesian inference.