A Classification of First Order Differential Equations

TL;DR

This paper classifies first order differential equations over a differential field of characteristic zero with algebraically closed constants and explores the algebraic dependence among their solutions, extending previous research.

Contribution

It provides a comprehensive classification of first order differential equations and analyzes the algebraic dependence of their solutions, building on and extending prior work.

Findings

Classification of first order differential equations over specified fields

Results on algebraic dependence of solutions

Generalizations of previous classifications

Abstract

Let $k$ be a differential field of characteristic zero with an algebraically closed field of constants. In this article, we provide a classification of first order differential equations over $k$ and study the algebraic dependence of solutions of a given first order differential equation. Our results generalize parts of the work of Noordman et al. (MR4378074) and complements the work of Freitag et al. (MR4506775).