On Rational Solutions of Systems of Linear Differential Equations

Let K be a field of characteristic zero and M(Y) = N a system of linear differential equations with coefficients in K(x). We propose a new algorithm to compute the set of rational solutions of such a system. This algorithm does not require the use of cyclic vectors. It has been implemented in Maple V and it turns out to be faster than cyclic vector computations. We show how one can use this algorithm to give a method to find the set of solutions with entries in log K(x)[ x ] of M(Y) = N.