Algorithms exist to find Llouvllltan 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 easyto-lmplcment simplified version of the Kovacic algorithm, based as much as possible on computation of rational solutions of linear differential equations.