Equivalence and integrability of second order linear ODEs

TL;DR

This paper analyzes second order linear ODEs with variable coefficients, constructing their Lie algebra of equivalence transformations, identifying invariants, and establishing criteria for their equivalence and integrability.

Contribution

It introduces a method to determine when second order linear ODEs are equivalent and integrable using Lie algebra invariants and differential invariants.

Findings

Constructed Lie algebra of equivalence transformations

Identified invariants and differential invariants

Formulated criteria for equation equivalence and integrability

Abstract

We consider a class of linear ODEs of second order with variable coefficients and construct its Lie algebra of Lie group of equivalence transformations. Further we find invariants and differential invariants of this Lie algebra and by using them we formulate criteria of equivalence of the equations under consideration. These criteria enable us to characterize some classes integrable in quadratures.