Note on Kovacicʹs Algorithm

Algorithms exist to find Liouvillian solutions of second order homogeneous linear differential equations (Kovacic, 1986, Singer and Ulmer, 1993b). In this paper, we show how, by carefully combining the techniques of those algorithms, one can find the Liouvillian solutions of an irreducible second order linear differential equation by computing only rational solutions of some associated linear differential equations. The result is an easy-to-implement simplified version of the Kovacicalgorithim, based as much on the computation of rational solutions of linear differential equations.