Let $e$ be a unit vector in $\mathbb R^d$, $d\geq 3$. Consider the function $\Gamma(x):=\frac {1}{|x-e|^{d-2}}$, which is the (translated) fundamental solution to the Laplacian on $\mathbb R^d$. Let $Z_n(x)$, $n\geq 2$ be the degree $n$ homogeneous term in the Taylor expansion of $\Gamma(x)$ at $x=0$. Show that $\nabla Z_n(x)$ vanishes at the origin but not elsewhere.
The problem is found in Wolff's article: Note on counterexamples in strong unique continuation problems
http://www.ams.org/journals/proc/1992-114-02/S0002-9939-1992-1014648-2/S0002-9939-1992-1014648-2.pdf
His arguments were "Since $Z_n(\frac {x}{|x|})$ is a solution to a second order ODE, $\nabla Z_n$ vanishes only at 0." I got completely lost in this sentence.