= Solution
Substitution of $y=1$ into the <general solution> gives
$$
\frac{x^2}{2}+F(x^2+2)=x^2.
$$
Thus $F(s)=(s-2)/2$ for $s\geq2$. Choosing this same formula everywhere gives the <polynomial> solution
$$
\boxed{u(x,y)=\frac12x^2y^2+\frac12x^2+y^2-1}.
$$
It directly satisfies the differential equation and $u(x,1)=x^2$.
\b[The <boundary condition> does not determine a unique solution on the entire <plane>.] Every characteristic <ellipse> $x^2+2y^2<2$ misses the data line $y=1$. More generally, $F$ is arbitrary for $0\leq s<2$, subject to matching the chosen regularity at $s=2$ and at the origin. The displayed <polynomial> is one global extension; it is forced on $x^2+2y^2\geq2$. A smooth <function> supported strictly inside $0<s<2$ may be added to $F$ without changing the boundary data.
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