On Liouvillian solutions of linear differential equations of order 4 and 5

Abstract

In this paper we give the minimal possible degrees of an algebraic solution of the Riccati equation associated to an irreducible linear homogeneous differential equation of order 4 and 5, following the method used for the order 3 ([14]). We show that the algebraic degree of such a solution is bounded by 120 for the order 4 and by 55 for the order 5. With the important work done by Hessinger in [5] and the Tables computed in the sequel this leads to an algorithm to find Galois group and liouvillian solutions of equations of order 4.