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Find the general solution of the partial differential equation

$𝑢𝑢_{xy} + 𝑢_𝑥𝑢_𝑦 = 0$.

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?

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Hint:

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

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

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