Differential transcendence of solutions of systems of linear differential equations based on total reduction of the system

Abstract

In this paper we consider total reduction of the nonhomogeneous linear system of operator equations with constant coefficients and commuting operators. The totally reduced system obtained in this manner is completely decoupled. All equations of the system differ only in the variables and in the nonhomogeneous terms. The homogeneous parts are obtained using the generalized characteristic polynomial of the system matrix. We also indicate how this technique may be used to examine differential transcendence of the solution of the linear system of the differential equations with constant coefficients over the complex field and meromorphic free terms.

Linear Systems of Differential Equations in Arrowhead Form

“…In [12] and [11] we have considered a partial and a total reduction of linear systems of operator equations with the system matrix in the companion form. Papers [12,11,10] and [13] expand our research to non-homogeneous linear systems of operator equations involving more than one operator.…”