On one criterion of expressibility of functions of a system of linear differential equations with constant coefficients in the form of linear combinations of derivatives of one function included in this system

In this paper, we study the problem of expressibility of all functions x1(t), x2(t), . . . , xn(t) included in a given homogeneous system of linear differential equations with constant coefficients x′(t) = A·x(t), in the form of linear combinations of derivatives of only one unknown function xк(t) included in this system. A simple criterion is found for the expressibility of all functions of the system x′(t) = A·x(t), in the form of linear combinations of derivatives xк(t), and its correctness is proved. Based on the proven criterion, an appropriate algorithm was developed and its correctness was substantiated.