The article is devoted to the question of reducibility systems of ordinary differential equations, which originates from Lyapunov's classical work [1]. The idea of this research was developed by N.P. Erugin in [2]. An attractive direction in the theory of reducibility is the problem of reducibility linear systems with oscillatory coefficients, in particular, in the periodic case, we have the well-known Floquet theory [3].
It should be noted that the question of reducibility linear systems with variable coefficients to systems with constant coefficients is of particular interest.
In this research, this question is posed from a general point of view. If two linear systems with variable coefficients based on a non-singular linear replacement are reducible to each other, that is, mutually reducible, then the problem arises of clarifying the relationship that exists between these systems and their transformations.
The purpose of this article is to study this problem in some special cases and to establish a connection with known results by reducibility of systems of ordinary differential equations.
In connection with the solution of this problem, the statement of the question is given in the note. The question of the solvability of this problem is considered. The mutual reducibility of linear systems with periodic coefficients is investigated. A connection is established between the question of mutual reducibility and the Floquet theory, and the question of mutual reducibility of systems is investigated using the Lyapunov matrix, known from the theory of stability of motions.
In conclusion, it is noted that the results obtained in this work can be generalized in the case of linear systems of
partial differential equations considered in [4-10].