This is the problem from Evans PDE Chapter 3.
Suppose that the formula $G(x,z,a) = 0$ implicitly defines the function $z = u(x,a)$, where $x,a \in \mathbb R^n$. Assume further that we can eliminate the variables $a$ from the identities $$\begin{cases} G(x,u,a) = 0 \\ G_{x_i}(x,u,a) + G_z(x,u,a)u_{x_i}= 0 \ (i=1,...,n)\end{cases}$$ to solve $u=u(x)$.
a) Find a PDE that u solves if $G = \sum_{i=1}^na_ix_i^2+z^3$.
b) What is the PDE characterizing all spheres in $\mathbb R^{n+1}$ with unit radius and center in $\mathbb R^n\times\{z=0\}$?
For part a, I use $G = \sum_{i=1}^na_ix_i^2+u^3=0$ and differentiating with respect to $x_i$, we have $2a_ix_i + 3u^2D_{x_i}u = 0.$ Hence $a_ix_i=-\frac{3}{2}u^2D_{x_i}u$ and we can rewrite $G = \sum_{i=1}^na_ix_i^2+u^3=-\frac{3}{2}u^2\cdot x\cdot D_{x}u+u^3=0$, which the desired PDE. Is this correct and how to approach part b?
For what I understand, you are doing it right in part a). I got almost the same: $G = \sum_{i=1}^na_ix_i^2+u^3=-3u^2\cdot x\cdot D_{x}u+u^3=0$. I think you forgot the $G_z$ part, correct me if I am wrong.
For part b) you should take the formula for an $n+1$-dimensional sphere $$\sum_{i=1}^n(x_i-a_i)^2=r^2$$ and for the $n+1$ term put $(u-0)^2$ (because $z=u$). And then use the same identities for $G$ as in part a).