Searching for Laurent Solutions of Truncated Systems of Linear Differential Equations with the Use of EG-Eliminations

Laurent solutions of systems of linear ordinary differential equations with truncated power series as coefficients are considered. The Laurent series in the solutions are also truncated. As a means for constructing such solutions, induced recurrent systems are used; earlier, an algorithm for the case when the induced recurrent system has a nonsingular leading matrix was proposed. For the series in solutions, this algorithm finds the maximum possible number of terms that are invariant with respect to any prolongation of the truncated coefficients of the original system. Results on extending the applicability of the earlier proposed algorithm to the case when the leading matrix is singular using the EG-elimination algorithm as an auxiliary tool. An implementation of the proposed algorithm in the form of a Maple procedure is given and examples of its use are presented.