 Solution

Substitution of $y=1$ into the general solution givesThus $F(s)=(s-2)/2$ for $s\ge 2$. Choosing this same formula everywhere gives the polynomial solutionIt directly satisfies the differential equation and $u(x,1)=x^2$.

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 $0\le s<2$, subject to matching the chosen regularity at $s=2$ and at the origin. The displayed polynomial is one global extension; it is forced on $x^2+2y^2\ge 2$. A smooth function supported strictly inside $0<s<2$ may be added to $F$ without changing the boundary data.