= Solution
Write $u(x,y)=F(\phi(x,y))$ for any differentiable $F$. The <multivariable chain rule> gives $u_x=2xF'(\phi)$ and $u_y=4yF'(\phi)$. Consequently
$$
2yu_x-xu_y=4xyF'(\phi)-4xyF'(\phi)=0.
$$
Thus $\boxed{u=F(x^2+2y^2)}$ 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>.
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