Consider the equation, u_x+2xy^2u_y=0 on D = \mathbb{R}^2 - \{ 0 \} \times \mathbb{R}.

Say that y solves the equation y' = 2xy^2 on (-\infty,0)\cup (0,\infty).
Then it must mean y=0 or y=\tfrac{1}{C - x^2}.

By the method of charachteristics it means u(x,y) = f(x^2 + \tfrac{1}{y}) where f is some differenciable function.

However, this is just a solution D.
Is there a non-trivial solution on \mathbb{R}^2?