I would like to solve an equation of the form
$$\varphi_t(t, s) + \alpha(s) \varphi(t, s - \beta(t)) = 0$$
but cannot find a single example of how to do so. Maybe I just don't have the right search terms. Any insights are greatly appreciated.
My attempt: Separation of variables is not a sure solution in the delay case, but it would be trivial if we assumed $\varphi(t, s) = \varphi_1(t) e^{-s}$, for example, since it turns the additive term into a multiplicative one. This doesn't make any sense once substituted, however, because of the $s$-dependence of $\alpha(s)$ — we end up forcing $\varphi_1(t)$ to depend on $s$, which can't be the case because of the initial assumptions.