Equivalence of differential equations of order one

TL;DR

This paper introduces notions of equivalence for first-order differential equations, explores their properties, and provides algorithms for testing strict equivalence and algebraic solutions, especially for autonomous equations and those with higher genus algebraic curves.

Contribution

It formalizes the concept of strict equivalence for first-order differential equations and develops algorithms to test this equivalence and the existence of algebraic solutions.

Findings

Strict equivalence can be tested algorithmically for equations with higher genus curves.

Testing strict equivalence for autonomous equations is algorithmically feasible.

The paper provides methods to determine algebraic solutions for certain classes of equations.

Abstract

The notions of equivalence and strict equivalence for order one differential equations are introduced. The more explicit notion of strict equivalence is applied to examples and questions concerning autonomous equations and equations having the Painleve property. The order one equation determines an algebraic curve. If this curve has genus zero or one, then it is difficult to verify strict equivalence. However, for higher genus strict equivalence can be tested by an algorithm sketched in the text. For autonomous equations, testing strict equivalence and the existence of algebraic solutions are shown to be algorithmic.