<s>[INST] What is general solution of the PDE $(x^2+y^2)u_x+2xyu_y=-u^2$? What is the general solution of the following PDE?

$$(x^2+y^2)u_x+2xyu_y=-u^2.$$

If we write the characteristic equation
$$\frac{dx}{x^2+y^2}=\frac{dy}{2xy}=\frac{-du}{u^2}$$
then, we find $\frac{d(x+y)}{(x+y)^2}=\frac{-du}{u^2}$ and we have $c_1=1/u+1/(x+y)$. 
And then?
Best regards.
 [/INST] HINT :
$$\frac{dx}{x^2+y^2}=\frac{dy}{2xy}=\frac{dx+dy}{(x+y)^2}=\frac{dx-dy}{(x-y)^2}
$$
$$\frac{1}{x+y}-\frac{1}{x-y}=c_2$$
With the first characteristic equation that you alredy found, this second equation (or any other equivalent equation) allows to express the general solution.
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