An implementation of Kovacic's algorithm for solving second order linear homogeneous differential equations

Abstract

Kovacic [3] has given an algorithm for the closed form solution of differential equations of the form ay" + by' + cy &equil; 0, where a, b, and c are rational functions with complex coefficients of the independent variable x. The algorithm provides a Liouvillian solution (i.e. one that can be expressed in terms of integrals, exponentials and algebraic functions) or reports that no such solution exists. In this note a version of Kovacic's algorithm is described. This version has been implemented in MACSYMA and tested successfully on examples in Boyce and DiPrima [1], Kamke [2], and Kovacic [3]. Modifications to the algorithm have been made to minimize the amount of code needed and to avoid the complete factorization of a polynomial called for. In Section 2 these issues are discussed and in Section 3 the author's current version of the algorithm is described.