Finding the general solution of $u u_{xy} - u_x u_y = 0,$

Question

In the book of Berg, at page 4, at the end of the introduction, as an exercise, it is asked to find the general solution of

$$u u_{xy} - u_x u_y = 0,$$

however, considering the fact that the book haven't showed any method or anything at all, how can we find the general solution of this PDE ?

Answer 1

\begin{align} uu_{xy}-u_xu_y &= 0 \\ \frac{u_xu_{xy}-u_xu_y}{u^2}&= 0 \\ \left(\frac{u_{y}}{u}\right)_x &= 0 \\ \frac{u_{y}}{u} &=h(y) \\ e^{-\int_0^y h(v) dv}u_{y} &=e^{-\int_0^y h(v) dv}h(y) u \\ (e^{-\int_0^y h(v) dv}u)_y &= 0\\ u &= g(x)e^{\int_0^y h(v) dv}=g(x)j(y)\\ \end{align}

Answer 2

Any solution of the form $u(x,y) = f(x)g(y)$ is valid, since

$$ uu_{xy} - u_xu_y = fg\cdot \frac{df}{dx}\frac{dg}{dy} - \frac{df}{dx}g\cdot f\frac{dg}{dy} = 0 $$

is always true