Generalized ODE reduction algorithm for bounded degree transformation

TL;DR

This paper presents an efficient Maple algorithm for transforming and reducing rational first-order ODEs, even when the numerator and denominator are not coprime, improving computational feasibility for large and complex equations.

Contribution

It introduces a generalized algorithm for ODE reduction that handles non-coprime cases with known degree bounds, enhancing previous methods' applicability and efficiency.

Findings

Algorithm effectively computes transformations for non-coprime cases

Implementation in Maple demonstrates practical utility

Reduces computational complexity for large ODEs

Abstract

The integrability problem of rational first-order ODEs $y^{\prime}=\frac{M(x,y)}{N(x,y)}$, where $M,N \in \mathbb{R}[x,y]$ is a long-term research focus in the area of dynamical systems, physics, etc. Although the computer algebra system such as Mathematica, Maple has developed standard algorithms to tackle its first integral expressed by Liouvillian or special function, this problem is quite difficult and the general method requires specifying a tight degree bound for the Darboux polynomial. Computing the bounded degree first integral, in general, is very expensive for a computer algebra system\cite{duarte2021efficient}\cite{cheze2020symbolic} and becomes impractical for ODE of large size. In \cite{huang2025algorithm}, we have proposed an algorithm to find the inverse of a local rational transformation $y \to \frac{A(x,y)}{B(x,y)}$ that transforms a rational ODE to a simpler and more tractable structure $y^{\prime}=\sum_{i=0}^nf_i(x)y^i$, whose integrability under linear transformation $\left\{x \to F(x),y \to P(x)y+Q(x)\right\}$ can be detected by Maple efficiently \cite{CHEBTERRAB2000204}\cite{cheb2000first}. In that paper we have also mentioned when $M(x,y),N(x,y)$ of the reducible structure are not coprime, canceling the common factors in $y$ will alter the structure which makes that algorithm fail. In this paper, we consider this issue. We conclude that with the exact tight degree bound for the polynomial $A(x,y)$ given, we have an efficient algorithm to compute such transformation and the reduced ODE for "quite a lot of" cases where $M,N$ are not coprime. We have also implemented this algorithm in Maple and the code is available in researchgate.