# Vestnik КRAUNC. Fiz.-Mat. nauki. 2023. vol. 43. no. 2. P. 20-30. ISSN 2079-6641

**MATHEMATICS**

https://doi.org/10.26117/2079-6641-2023-43-2-20-30

Research Article

Full text in Russian

MSC 34A30, 90C90

**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**

**D. N. Barotov¹^\ast, R. N. Barotov²**

¹Financial University under the Government of the Russian Federation, 109456, Moscow,4-th Veshnyakovsky Passage, 4, Russia

²Khujand state university named after academician Bobojon Gafurov, 683032, 735700, Khujand, Mavlonbekov ave., 1, Tajikistan

**Abstract.** 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^\prime\left(t\right) = A· x\left(t\right) 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^\prime\left(t\right) = A· x\left(t\right) under any initial condition x\left(t_0\right) = x_0 can be expressed as linear combinations of derivatives of only one given component x_k\left(t\right). 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^\prime\left(t\right) = A· x\left(t\right) for periodicity and in the study of linear systems for complete observability.

*Key words: homogeneous system of linear differential equations with constant coefficients, method for **reducing a system of linear equations to a single higher-order equation, expressibility criterion, algorithm*

Received: 28.03.2023; Revised: 29.06.2023; Accepted: 03.07.2023; First online: 07.07.2023

**For citation.** Barotov D. N., Barotov R. N. 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. Vestnik KRAUNC. Fiz.-mat. nauki. 2023, 43: 2, 20-30. EDN: KJHTVW. https://doi.org/10.26117/2079-6641-2023-43-2-20-30.

**Funding.** The work was carried out without the support of funds.

**Competing interests.** There are no conflicts of interest regarding authorship and publication.

**Contribution and Responsibility.** All authors contributed to this article. Authors are solely responsible for providing

the final version of the article in print. The final version of the manuscript was approved by all authors.

^\astCorrespondence: E-mail: dnbarotov@fa.ru

The content is published under the terms of the Creative Commons Attribution 4.0 International License

© Barotov D. N., Barotov R. N., 2023

© Institute of Cosmophysical Research and Radio Wave Propagation, 2023 (original layout, design, compilation)

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**Information about authors**

**Barotov Dostonjon Numonjonovich **– Senior Lecturer; Department of Data Analysis and Machine Learning, Financial University under the Government of the Russian Federation, Moscow, Russia, ORCID 0000-0001-5047-7710.

**Barotov Ruziboy Numonjonovich **– Doctoral Student; Department of Mathematical Analysis named after Professor A. Mukhsinov, Khujand state university named after academician Bobojon Gafurov, Khujand, Tajikistan, ORCID 0000-0003-3729-6143.