Formal Power Series Solutions of Algebraic Ordinary Differential Equations

In this paper, we consider nonlinear algebraic ordinary differential equations (AODEs) and study their formal power series solutions. Our method is inherited from Lemma 2.2 in [J. Denef and L. Lipshitz, Power series solutions of algebraic differential equations, Mathematische Annalen, 267(1984), 213-238] for expressing high order derivatives of a differential polynomial via their lower order ones. By a careful computation, we give an explicit formula for the expression. As an application, we give a method for determining the existence of a formal power series solution with given first coefficients. We define a class of certain differential polynomials in which our method works properly, which is called non-vanishing. A statistical investigation shows that many differential polynomials in the literature are non-vanishing.