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I want to solve this equation: $$ -2u_{x}\cdot u_{y}+u\cdot u_{xy}=k $$ where $k$ is a constant.

I only know about linear partial differential equation and I could not find many information about non linear PDEs. Someone know if there is a way to get a general solution? Any reference?

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$$ -2u_{x}\cdot u_{y}+u\cdot u_{xy}=k $$

HINT :

The change of function $\quad u(x,y)=\frac{1}{v(x,y)}\quad$ transforms the PDE to a much simpler form : $$v_{xy}=-k\:v^3$$ I doubt that a closed form exists to analytically express the general solution. It is better to consider some numerical methods.

If the boubdary conditions are explicitly defined, the question has to be reconsidered according to this complementary information.

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