Find the general solution $u(x,y)$ to $$x u_x + y u_y = 0$$
Attempted solution - The characteristic equation satisfy the ODE $dy/dx = y/x$. To solve the ODE, we separate variables: $dy/y = dx/x$; hence $\ln(y) = \ln(x)- C$, so that $$y = x \exp(-C)$$
I am a bit confused in finding the general solution for $u(x,y)$. I just want to know if I am on the right track or not. I haven't taken a formal ODE course but I have self-studied but not in a while. Any suggestions would be greatly appreciated.