On the representability of functions of the system 𝑥′(𝑡) = 𝐴 𝑥(𝑡)in the form of linear combinations of derivatives of one scalarfunction of the form 𝑦(𝑡) = ⟨𝑐, 𝑥(𝑡)⟩

In this paper, for a system of linear differential equations of the form 𝑥′(𝑡) =𝐴 · 𝑥(𝑡), we study the problem of the existence of a scalar function of the form 𝑦(𝑡) =( ) ( ) ⟨𝑐, 𝑥(𝑡)⟩ such that 𝑥1(𝑡), 𝑥2(𝑡), ..., 𝑥𝑛(𝑡) - all components of the solution vector of the system 𝑥′(𝑡) = · 𝑥(𝑡) for any initial condition 𝑥(𝑡0) = 𝑥0 belong to the linear hull𝑆𝑝⟨𝑐, 𝑥(𝑡)⟩, ⟨𝑐, 𝑥(𝑡)⟩′, ⟨𝑐, 𝑥(𝑡)⟩′′, ..., ⟨𝑐, 𝑥(𝑡)⟩(𝑛-1). An existence criterion was formulated𝑆𝑝⟨𝑐, 𝑥(𝑡)⟩, ⟨𝑐, 𝑥(𝑡)⟩′, ⟨𝑐, 𝑥(𝑡)⟩′′, ..., ⟨𝑐, 𝑥(𝑡)⟩(𝑛-1). An existence criterion was formulated 𝑆𝑝𝑆𝑝··and proved, complementing previously published results on the representability of all components of the system 𝑥′(𝑡) = 𝑥(𝑡) in the form of linear combinations of derivativesof one scalar function. It is also argued that when the representability criterion is met, the periodicity, almost periodicity, and boundedness of the solution vector 𝑥(𝑡) of the system𝑥′(𝑡) = · 𝑥(𝑡) follow only from the periodicity, almost periodicity, and boundedness of onefunction 𝑦(𝑡) = ⟨𝑐, 𝑥(𝑡)⟩, respectively.