Invariants of differential equations defined by vector fields

TL;DR

This paper identifies the most general transformations for differential equations defined by vector fields, finds their invariants up to second order, and characterizes classes of these equations using invariant functions.

Contribution

It provides a comprehensive analysis of invariance properties and classification methods for differential equations defined by arbitrary vector fields.

Findings

Determined the most general group of equivalence transformations.

Found all invariants and differential invariants up to second order.

Characterized classes of equations via invariant functions.

Abstract

We determine the most general group of equivalence transformations for a family of differential equations defined by an arbitrary vector field on a manifold. We also find all invariants and differential invariants for this group up to the second order. A result on the characterization of classes of these equations by the invariant functions is also given.