I am trying to solve a PDE of the form
$$\xi_t(t, x) + a(t) x^{f(t)} \xi(t, x) + b(t) x^{f(t)+1} \xi_x(t,x) + c(t) x^{f(t)+2} \xi_{xx}(t, x) = 0.$$
I have no idea if this is even possible. There seem to be some nice properties under a Mellin transform, but it turns into a delay differential equation unlike the examples I've seen, where $\varphi(t, s)$ is the Mellin transform of $\xi(t, x)$. For my particular case, the above equation becomes
$$\varphi_t(t, s) + g(s) \varphi(t, s + f(t)) = 0.$$
I've assumed that $\xi(t, x = \infty) = \xi_x(t, x = \infty) = 0$. This looks easier to solve, but I don't know where to go from here. Any help is greatly appreciated.
Update 1: Surprising no one, I took the transform incorrectly; this has now been fixed.
Update 2: After quite a bit of work, I was able to guess a solution to the above equation of the form
$$\varphi(t, s) = \varphi_1(s) \varphi_1(s + f(t)).$$
The partial delay differential equation then becomes an ordinary delay differential equation
$$g(s) \varphi_1(s - f(t)) + f^\prime(t) \varphi_1^\prime(s + f(t)) = 0$$
and, after the change of variables $q = s + f(t)$, we have
$$g(q - f(t)) \varphi_1(q - 2 f(t)) + f^\prime(t) \varphi_1^\prime(q) = 0.$$
This at least looked like something that can be solved for a chosen $f(t)$ and $g(s)$, so I went through the method of steps for a few different choices. I can confirm the solution both analytically and numerically, but... how do I inverse Mellin transform the method of steps solution (a piecewise function, typically a power series in $t$ and $s$ for my purposes) to get back to what I wanted in the first place?