Solving First Order Autonomous Algebraic Ordinary Differential Equations by Places

TL;DR

This paper introduces algorithms for computing and classifying formal power series solutions of first-order autonomous algebraic ODEs using the concept of places, including convergence proofs over complex numbers.

Contribution

It presents novel algorithms for characterizing initial values and computing solutions, with convergence guarantees over the complex field.

Findings

Algorithms successfully classify initial values and compute solutions.

All computed solutions are proven to converge in complex neighborhoods.

Provides a comprehensive framework for formal power series solutions of algebraic ODEs.

Abstract

Given a first-order autonomous algebraic ordinary differential equation, we present a method for computing formal power series solutions by means of places. We provide an algorithm for computing a full characterization of possible initial values, classified in terms of the number of distinct formal power series solutions extending them. In addition, if a particular initial value is given, we present a second algorithm that computes all the formal power series solutions, up to a suitable degree, corresponding to it. Furthermore, when the ground field is the field of the complex numbers, we prove that the computed formal power series solutions are all convergent in suitable neighborhoods.