  Several algorithms in computer algebra involve the computation of a power
series solution of a given ordinary differential equation. Over finite fields,
the problem is often lifted in an approximate $p$-adic setting to be
well-posed. This raises precision concerns: how much precision do we need on
the input to compute the output accurately? In the case of ordinary
differential equations with separation of variables, we make use of the recent
technique of differential precision to obtain optimal bounds on the stability
of the Newton iteration. The results apply, for example, to algorithms for
manipulating algebraic numbers over finite fields, for computing isogenies
between elliptic curves or for deterministically finding roots of polynomials
in finite fields. The new bounds lead to significant speedups in practice.
