Liouvillian and Algebraic Solutions of Second and Third Order Linear Differential Equations

Abstract

In this paper we show that the index of a 1-reducible subgroup of the differential Galois group of an ordinary homogeneous linear differential equation L( y) = 0 yields the best possible bound for the degree of the minimal polynomial of an algebraic solution of the Riccati equation associated to L( y) = 0. For an irreducible third order equation we show that this degree belongs to {3,6,9,21,36}. When the Galois group is a finite primitive group, we reformulate and generalize work of L. Fuchs to show how to compute the minimal polynomial of a solution instead of the minimal polynomial of the logarithmic derivative of a solution. These results lead to an effective algorithm to compute Liouvillian solutions of second and third order linear differential equations.