Integration of first-order ODEs by Jacobi fields

TL;DR

This paper introduces a novel class of vector fields for integrating first-order ODEs, based on geometric relations with Riemannian manifolds, expanding the methods beyond traditional Lie symmetries.

Contribution

It presents a new integration method using vector fields not limited to Lie symmetries, linked to Riemannian geometry, with practical examples and a connection to Schrödinger-type equations.

Findings

New vector fields enable integration of first-order ODEs.

Established a geometric relation between Riemannian manifolds and ODE integrability.

Provided examples illustrating the integration procedure.

Abstract

A new class of vector fields enabling the integration of first-order ordinary differential equations (ODEs) is introduced. These vector fields are not, in general, Lie point symmetries. The results are based on a relation between 2-dimensional Riemannian manifolds and the integrability of first-order ODEs, which was established in a previous work of the authors. An integration procedure is provided, together with several examples to illustrate it. A connection between integrating factors of first-order ODEs and Schr\"odinger-type equations is highlighted.