be a linear differential operator with coefficients in F, a finite algebraic extension of Q(x). We shall show that one can find, in a finite number of steps, a basis for the vector space of liouvillian solutions of L ( Y) = 0 (i.e., those solutions which can be built up from the rational functions by algebraic operations, taking exponentials and by integration; see Section 2 for a precise definition). In particular, we show how to decide if all solution of L(Y) = 0 are liouvillian or if there are any solutions which are liouvillian. Our algorithm in conjunction with the algorithm in [8], also allows one to determine the algebraic relationships among the liouvillian solutions of L(Y) = 0, and, in particular, determine if all solutions are algebraic (c.f. [111). For second order linear homogeneous equations over Q(x), the problem of determining if all solutions are algebraic functions was considered by Fuchs, Klein and Schwarz, but none of these mathematicians seems to have presented a complete decision procedure. Building on the work of Klein, Baldassarri and Dwork [21 have given such a procedure. Baldassarri [11 has extended this to consider linear homogeneous equations whose coefficients are algebraic functions. For third order equations, Painleve and Boulanger gave a procedure which, in effect, reduced the problem of finding algebraic solutions of L( Y) = 0 to the problem of effectively bounding the torsion of the jacobian variety of a given curve. A complete procedure for deciding if all solutions of an n-th order homo-