= Solution
The initial curve $y^3=9x$ is \b[$\eta=0$], with $\xi=2\sqrt x\in[1,2]$, so it is characteristic. The displacement data force $g(\xi)+h(0)=\xi$ and therefore $g'=1$ on that interval. Along the curve,
$$
u_y=\frac{\sqrt y}{2}\bigl(1-h'(0)\bigr),\qquad\sqrt y=3^{1/3}x^{1/6}.
$$
The derivative data in case (i) are compatible precisely when $h'(0)=-1$. They place no restriction on the rest of $h$. For example, for every real constant $C$,
$$
\boxed{u_C(x,y)=\frac23y^{3/2}+C\left(\sqrt x-\frac{y^{3/2}}3\right)^2}
$$
solves the equation and has both prescribed data on the curve. Choosing $C=0$ and $C=1$ exhibits \b[more than one solution] directly. The initial curve follows one characteristic family, so its data do not determine the other wave family. This is the compatible case of <compatibility of Cauchy data on a characteristic curve>.
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