Computing the differential Galois group of a parameterized second-order linear differential equation

Abstract

We develop algorithms to compute the differential Galois group G associated to a parameterized second-order homogeneous linear differential equation of the form [EQUATION] where the coefficients r₁, r₀ ∈ F(x) are rational functions in x with coefficients in a partial differential field F of characteristic zero. This work relies on earlier procedures developed by Dreyfus and by the present author to compute G when r₁ = 0. By reinterpreting a classical change-of-variables procedure in Galois-theoretic terms, we complete these algorithms to compute G with no restrictions on r₁.