= Solution
On $y=1$, the transformed variables satisfy $s=1/3$ and $1/2\leq r\leq1$. Differentiating the displacement data tangentially and using $u_y=1$ gives
$$
g'(r+1/3)+h'(r-1/3)=2,\qquad \tfrac12\bigl(g'(r+1/3)-h'(r-1/3)\bigr)=1.
$$
Hence $g'=2$ on $[5/6,4/3]$, $h'=0$ on $[1/6,2/3]$, and the additive constants have sum $-2/3$. The uniquely determined solution is therefore
$$
\boxed{u(x,y)=2\sqrt x+\frac23(y^{3/2}-1).}
$$
It is fixed by the data exactly where both needed characteristic arguments belong to their known intervals:
$$
\boxed{\frac56\leq\sqrt x+\frac{y^{3/2}}3\leq\frac43,\qquad\frac16\leq\sqrt x-\frac{y^{3/2}}3\leq\frac23.}
$$
Equivalently, in the $(r,s)$ plane this is the diamond $|r-3/4|+|s-1/3|\leq1/4$. In the original coordinates it is bounded by four curved <characteristic curves>. It lies wholly in $x,y>0$. The displayed formula supplies one smooth extension beyond the diamond too, but the specified finite data segment does not determine that extension: one may change $g$ or $h$ outside its known argument interval. This is the <characteristic-coordinate domain of dependence>.
\Image[/past-exam-of-the-differential-equations-1-course-of-the-university-of-oxford/2017-characteristic-domain.png]
{title=The finite Cauchy-data segment determines a characteristic diamond in transformed coordinates}
{description=The left panel shows its curved image in the original quadrant; the right panel shows the straight-sided diamond after the change of variables. Shading includes both forward and backward determination from the data segment.}
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