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

Say that $\displaystyle y$ solves the equation $\displaystyle y' = 2xy^2$ on $\displaystyle (-\infty,0)\cup (0,\infty)$.

Then it must mean $\displaystyle y=0$ or $\displaystyle y=\tfrac{1}{C - x^2}$.

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

However, this is just a solution $\displaystyle D$.

Is there a non-trivial solution on $\displaystyle \mathbb{R}^2$?