On Existence and Uniqueness of Formal Power Series Solutions of Algebraic Ordinary Differential Equations

TL;DR

This paper introduces a computational method to determine the existence, uniqueness, and structure of formal power series solutions for algebraic ordinary differential equations, under certain regularity conditions.

Contribution

It provides a novel approach to identify when truncated power series solutions can be extended and characterizes the algebraic structure of all solutions.

Findings

Method determines extendability of truncated solutions

Conditions for existence and uniqueness are established

Algebraic structure of solution set is described

Abstract

Given an algebraic ordinary differential equation (AODE), we propose a computational method to determine when a truncated power series can be extended to a formal power series solution. If a certain regularity condition on the given AODE or on the initial values is fulfilled, we compute all of the solutions. Moreover, when the existence is confirmed, we present the algebraic structure of the set of all formal power series solutions.