  We consider a finite \'etale morphism $f:Y \to X$ of quasi-smooth Berkovich
curves over a complete nonarchimedean non-trivially valued field $k$, assumed
algebraically closed and of characteristic 0, and a skeleton
$\Gamma_f=(\Gamma_Y,\Gamma_X)$
  of the morphism $f$. We prove that $\Gamma_f$ radializes $f$ if and only if
$\Gamma_X$ controls the pushforward of the constant $p$-adic differential
equation $f_*(\mathcal{O}_Y,d_Y)$.
  Furthermore, when $f$ is a finite \'etale morphism of open unit discs, we
prove that $f$ is radial if and only if the number of preimages of a point
$x\in X$, counted without multiplicity, only depends on the radius of the point
$x$.
