Formal Power Series Solutions of Algebraic Ordinary Differential Equations
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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.
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