On the Equivalence Problem of Generalized Abel ODEs under the Action of the Linear Transformations Pseudogroup

TL;DR

This paper determines the precise conditions under which two generalized Abel differential equations are locally equivalent through a specific pseudogroup of linear transformations, using differential invariants.

Contribution

It establishes necessary and sufficient conditions for local equivalence of generalized Abel ODEs under a particular pseudogroup, formulated via differential invariants.

Findings

Derived explicit differential invariants for the pseudogroup

Provided criteria for local equivalence of generalized Abel equations

Enhanced understanding of symmetry transformations in differential equations

Abstract

In the present paper we establish the necessary and sufficient conditions for two generalized Abel differential equations to be locally equivalent under the action of the pseudogroup of linear transformations of the form $\{x\mapsto f(x),~ y\mapsto g(x)\cdot y + h(x)\}$. These conditions are formulated in terms of differential invariants.