Is there a general solution for first-order partial differential equations of the form
$$m(x) \partial_x f(x,y) = n(y) \partial_y f(x,y)$$
for given $m(x),n(y)$ and reasonable boundary conditions (e.g. $f(x,0)=0$ etc.)?
Change the variables in $x$ and $y$, i.e., $\tilde x =\tilde x(x)$ and $\tilde y = \tilde y(y)$ to make $m(x) = n(y) = 1$. Then equation can be written as
$ \partial_{\tilde x}f =\partial_{\tilde y}f. $
The equation above is equivalent to that $f= f(\tilde x+ \tilde y)$.