I would start from the beginning.
$$\frac{u_{xy}}{u_x} = \frac{u_x u_y}{u_x} \Rightarrow \frac{u_{xy}}{u_x} = u_y$$
Integrate both sides,
$$\ln{u_x} = u + f(x)$$
$$u_x = e^{u+f(x)} = e^u \cdot g(x)$$
$$\frac{\partial u}{\partial x} = g(x)e^u$$
$$\frac{\partial u}{e^u} = g(x)\partial x$$
Integrate both sides,
$$-e^{-u} = G(x) + h(y)$$
$$u(x,y) = -\ln (-G(x) - h(y))$$
OK. V.I.
