= Elliptic characteristics of a planar transport equation
For the <linear partial differential equation> $2y u_x-xu_y=0$, the <method of characteristics> gives $\dot x=2y$, $\dot y=-x$, $\dot u=0$. Thus $\phi=x^2+2y^2$ is constant on each <characteristic curve>, whose nonzero level sets are <ellipses>. On a domain containing entire such <ellipses>, the solutions have the form $u=F(\phi)$, subject to the desired regularity at the origin.
For the inhomogeneous equation $2yu_x-xu_y=xy(2y^2-x^2)$, a <particular solution> is $u_p=x^2y^2/2$, so the <general solution> is $u=x^2y^2/2+F(x^2+2y^2)$. Prescribing $u(x,1)=x^2$ fixes $F(s)=(s-2)/2$ only for $s\geq2$. The inner <ellipses> $s<2$ never intersect that data line, so the boundary values leave their solution values undetermined. Smooth extensions must match the prescribed $F$ at $s=2$; choosing the same linear formula everywhere gives one polynomial extension.
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