Simultaneously row- and column-reduced higher-order linear differential systems

Abstract

In this paper, we investigate the local analysis of systems of linear differential-algebraic equations (DAEs) and second-order linear differential systems. In the first part of the paper, we show how one can transform an input linear DAE into a reduced form that allows for the decoupling of the differential and algebraic components of the system. Classification of singularities of linear DAEs are defined and discussed. In the second part of the paper, we extend this approach to second-order linear differential systems and discuss two applications: the classification of singularities and the computation of regular solutions. The present paper is the first step towards a generalisation of the formal reduction of first-order ODEs to higher-order systems. Our algorithm has been implemented in the computer algebra system Maple as part of the ISOLDE package.