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.