Given an operator and boundary conditions, there often exist eigenfunctions which allow for Fourier summation solutions.
Is there a similar way to solve initial value problems, for example,
$$ \partial_t^2 \psi = s(t) \\ \text{s.t. } \psi(0) = 0 \\ \psi'(0) = 0 $$
Can I find functions that are eigen-functions satisfying the initial conditions? They can be eigen-functions under any inner product.
This isn't an answer, just information in case other students find this confusing like me.
$$ \partial_t^2 \psi = \lambda \psi $$ can be Laplace transformed as, (with some algebra done)
$$ (s^2 - \lambda)\tilde{\psi} = s \psi(0) + \psi'(0) \\ \tilde{\psi} = \frac{s \psi(0) + \psi'(0)}{s^2 - \lambda} $$
Thus there are clearly conditions of the initial conditions for non-trivial solutions.
What I personally found to satisfy my search is that I can form Green's functions via the Laplace transform by solving,
$$ \hat{L} \psi = \delta(t - t_0) $$
and using the Laplace transform properties to handle the initial conditions.