I want to solve the following pde $$\left\{\begin{array}{cc} u_xu_y=u \mbox{ on $\Omega:=\{(x,y)|x>0\}$} \\ u(0,y)=y^2 \end{array}\right.$$ I supposed that $u$ was a polynomial of two variables of degree 2 and I found that $u(x,y)=y^2+\frac{1}{2}xy+\frac{1}{16}x^2$ is a solution of the pde. I want to know if there is way to solve this pde without guessing what the solution looks like.
I thought of using the characteristic method but I think there's missing a condition on the first partial derivatives of $u$ to use this method.
Thanks in advance for your help.