Formal Power Series Solutions of First Order Autonomous Algebraic Ordinary Differential Equations
نویسندگان
چکیده
Given a first order autonomous algebraic ordinary differential equation, we present a method to compute all formal series solutions. Furthermore, when the ground field is the field of the complex numbers, the computed formal power series solutions are indeed convergent in suitable neighborhoods. keywordAlgebraic autonomous differential equation, algebraic curve, local parametrization, formal power series solution.
منابع مشابه
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 on...
متن کاملLaurent Series Solutions of Algebraic Ordinary Differential Equations
This paper concerns Laurent series solutions of algebraic ordinary differential equations (AODEs). We first present several approaches to compute formal power series solutions of a given AODE. Then we determine a bound for orders of its Laurent series solutions. Using the order bound, one can transform a given AODE into a new one whose Laurent series solutions are only formal power series. The ...
متن کاملA solution method for autonomous first-order algebraic partial differential equations in several variables
In this paper we present a procedure for solving first-order autonomous algebraic partial differential equations in an arbitrary number of variables. The method uses rational parametrizations of algebraic (hyper)surfaces and generalizes a similar procedure for first-order autonomous ordinary differential equations. In particular we are interested in rational solutions and present certain classe...
متن کاملA solution method for autonomous first-order algebraic partial differential equations
In this paper we present a procedure for solving first-order autonomous algebraic partial differential equations. The method uses rational parametrizations of algebraic surfaces and generalizes a similar procedure for first-order autonomous ordinary differential equations. In particular we are interested in rational solutions and present certain classes in which such solutions exist. However, t...
متن کاملOn the complexity of solving ordinary differential equations in terms of Puiseux series
We prove that the binary complexity of solving ordinary polynomial differential equations in terms of Puiseux series is single exponential in the number of terms in the series. Such a bound was given by Grigoriev [10] for Riccatti differential polynomials associated to ordinary linear differential operators. In this paper, we get the same bound for arbitrary differential polynomials. The algori...
متن کامل