 Solution

Write $u(x,y)=F(\varphi (x,y))$ for any differentiable $F$. The multivariable chain rule gives $u_x=2x{F}^{\prime }(\varphi )$ and $u_y=4y{F}^{\prime }(\varphi )$. ConsequentlyThus $\boxed{u=F(x^2+2y^2)}$ solves the homogeneous equation. The method of characteristics also explains completeness on the full plane: the characteristic curves are the connected ellipses $x^2+2y^2=\text{constant}$, and $u$ is constant on each one. These are the elliptic characteristics of a planar transport equation.