Puiseux Series and Algebraic Solutions of First Order Autonomous AODEs -- A MAPLE Package

TL;DR

This paper introduces a MAPLE package that implements algorithms for computing all formal Puiseux series and algebraic solutions of autonomous first order algebraic differential equations, ensuring solution existence, uniqueness, and comprehensive solution description.

Contribution

The paper presents a new MAPLE package that automates the computation of all formal Puiseux series and algebraic solutions for autonomous first order algebraic ODEs, including singular solutions.

Findings

All solutions can be uniquely represented by truncations or minimal polynomials.

The package computes both generic and singular solutions effectively.

The method reduces equations using local parametrizations and degree bounds.

Abstract

There exist several methods for computing exact solutions of algebraic differential equations. Most of the methods, however, do not ensure existence and uniqueness of the solutions and might fail after several steps, or are restricted to linear equations. The authors have presented in previous works a method to overcome this problem for autonomous first order algebraic ordinary differential equations and formal Puiseux series solutions and algebraic solutions. In the first case, all solutions can uniquely be represented by a sufficiently large truncation and in the latter case by its minimal polynomial. The main contribution of this paper is the implementation, in a MAPLE-package named FirstOrderSolve, of the algorithmic ideas presented therein. More precisely, all formal Puiseux series and algebraic solutions, including the generic and singular solutions, are computed and described uniquely. The computation strategy is to reduce the given differential equation to a simpler one by using local parametrizations and the already known degree bounds.