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 \[ \tfrac{\partial^2}{\partial x^2} Y + r_1 \tfrac{\partial}{\partial x} Y + r_0 Y = 0, \] where the coefficients $r_1,r_0\in F(x)$ are rational functions in $x$ with coefficients in a partial differential field $F$ of characteristic zero. Our work relies on the procedure developed by Dreyfus to compute $G$ under the assumption that $r_1=0$. We show how to complete this procedure to cover the cases where $r_1\ne 0$, by reinterpreting a classical change of variables procedure in Galois-theoretic terms.

14 pages

References in corpus (3)

• Galois Theory of Parameterized Differential Equations and Linear Differential Algebraic Groups
• Computing the differential Galois group of a one-parameter family of second order linear differential equations
• Computation of the unipotent radical of the differential Galois group for a parameterized second-order linear differential equation

Cited by in corpus (3)

• Calculating differential Galois groups of parametrized differential equations, with applications to hypertranscendence
• Computing differential Galois groups of second-order linear $q$-difference equations
• Computation of the unipotent radical of the differential Galois group for a parameterized second-order linear differential equation