Find the general solution of the pde $_{xy} + __ = 0$

Question

Find the general solution of the partial differential equation

$_{xy} + __ = 0$.

This is a second order quasilinear equation, it cannot be solved using the method of characteristics. Does it have another way to find the general solution?

Answer 1

Hint:

$$\left(uu_{x}\right)_{y}=uu_{xy}+u_{x}u_{y}$$

$$\left(\dfrac{1}{2}u^{2}\right)_{x}=uu_{x}$$