## Solution on the Whole Plane

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 $\mathbb{R}^2$?