Computing the Galois group of some parameterized linear differential equation of order two

TL;DR

This paper extends Kovacic's algorithm to compute the differential Galois group of certain second order parameterized linear differential equations, providing criteria for integrability when no Liouvillian solutions exist.

Contribution

It introduces an extension of Kovacic's algorithm for parameterized equations and establishes conditions for integrability in the absence of Liouvillian solutions.

Findings

Extended Kovacic's algorithm for parameterized equations

Necessary and sufficient conditions for system integrability

Examples demonstrating the computation process

Abstract

We extend Kovacic's algorithm to compute the differential Galois group of some second order parameterized linear differential equation. In the case where no Liouvillian solutions could be found, we give a necessary and sufficient condition for the integrability of the system. We give various examples of computation.