# Finding wave solution for $\frac{\partial^2 f(x,t)}{\partial t^2} - g(t) \frac{\partial f(x,t)}{\partial t} - \frac{\partial^2 f(x,t)}{\partial x^2}$

I want to solve \begin{align} \frac{\partial^2 f(x,t)}{\partial t^2} - g(t) \frac{\partial f(x,t)}{\partial t} - \frac{\partial^2 f(x,t)}{\partial x^2} =0 \end{align} I am trying to find the general solution to this equation can you give me some nice ansatz for this?

For example, I know the solution for $$g(t)=0$$, In this case, \begin{align} f(x,t) = e^{ i (wt - kx)} f_0 \end{align} with $$w^2 = k^2$$.

Also for $$g(t) = g$$, constant I also have solution, For example, Fundamental solution for 1D nonhomogeneous wave equation

How about the general case, $$g=g(t)$$?. Is there any systematic way to solve this equation with the form of wave equation? [ I mean which satisfies some certain dispersion relation]

• Note, there's no uniqueness theorem for partial differential equations, so there's no general solution in the sense that it exists for ODEs, you should still specify what kind of solution you are looking for Commented Apr 17, 2019 at 9:10
• @Yurij S, What I want is the solution with "Dispersion relation". For example I want the solution form as $f(x,t) = e^{someting related to g(t) t} e^{i(wt-kx)} f_0$ with dispersino relation Commented Apr 17, 2019 at 9:12
• You are looking for a solution in the form $u(t) v(x)$, so first you can separate the variables. Then you get an ODE for $u(t)$ which can be solved by the usual methods Commented Apr 17, 2019 at 9:20