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.)?



Solution to simple first-order partial differential equations
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 $f= f(\tilde x+ \tilde y)$.


