On the refinement of the method of reducing a system of linear differential equations to a single higher-order equation, which makes it possible to find a general solution to the original

The theory of differential equations is currently an exceptionally content-rich, rapidly developing branch of mathematics, closely related to other areas of mathematics and its applications. When studying specific differential equations that arise in the process of solving physical problems, methods are created that have great generality and are applied to a wide range of mathematical problems. The problem of integrating differential equations with constant coefficients had a great influence on the development of linear algebra. At present, the problem of solving a system of linear ordinary differential equations with constant coefficients $x'(t)=A\cdot x(t)$ is one of the most important problems in both the theory of ordinary differential equations and linear algebra. One of the most well-known methods for solving a system of linear ordinary differential equations with constant coefficients is the method of reducing a system of linear equations to a single higher-order equation, which makes it possible to find solutions to the original system in the form of linear combinations of derivatives of only one function. In this paper, we study the following problem: for which matrices $A$ the components of the system $x'(t)=A\cdot x(t)$ under any initial condition $x(t_0)=x_0$ can be expressed as linear combinations of derivatives of only one given component $x_k(t)$. A new simple expressibility criterion is formulated, and its correctness is proved in detail. The result obtained can also be applied in the study of solutions of the system $x'(t)=A\cdot x(t)$ for periodicity and in the study of linear systems for complete observability.