Some types of differential equations over a field of finite characteristic (Q1311158)

Let \(K\) be a field of characteristic \(p>0\), \(B\) be the algebra of truncated polynomials over the field \(K\) \((B\subseteq K[x]/ (x^ p))\), \(f\in K[y]\) and \(\deg(f)<p\). The author considers the Cauchy problem \(y'= f(y)\), \(y(0)=c\) (\(c\in K\)). It follows from his previous paper [see ``The Cauchy problem in the divided power algebra'', VINITI (Moskva), Dep. No. 8830-B88] that the problem possesses a unique solution in a formal completion \(\widetilde{B}\) of \(B\). The present paper contains a proof of a theorem which describes when such a solution lies in \(B\).