 Solution

The initial curve $y^3=9x$ is $\eta =0$, with $\xi =2\sqrt{x}\in [1,2]$, so it is characteristic. The displacement data force $g(\xi )+h(0)=\xi$ and therefore $g^{\prime }=1$ on that interval. Along the curve,The derivative data in case (i) are compatible precisely when $h^{\prime }(0)=-1$. They place no restriction on the rest of $h$. For example, for every real constant $C$,solves the equation and has both prescribed data on the curve. Choosing $C=0$ and $C=1$ exhibits 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.