DOI: https://doi.org/10.26089/NumMet.v24r319

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

Authors

  • Dostonjon N. Barotov
  • Ruziboy N. Barotov

Keywords:

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

Abstract

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.

Downloads

Published

2023-07-05

Issue

Vol. 24 (2023): Issue 3.

Section

Methods and algorithms of computational mathematics and their applications

Author Biographies

Dostonjon N. Barotov

Financial University under the Government of the Russian Federation
Department of Data Analysis and Machine Learning
• Senior Lecturer

Ruziboy N. Barotov

Khujand state university named after academician Bobojon Gafurov
Department of Mathematical Analysis named after Professor A. Mukhsinov
• Doctoral Student

References

  1. O. A. Oleinik, “The Role of the Theory of Differential Equations in Modern Mathematics and Its Applications,” Soros Educational Journal, No. 4, 114-121 (1996).
  2. V. V. Kalinin, Ordinary Differential Equations (Gubkin Univ. Press, Moscow, 2017) [in Russian].
  3. F. R. Gantmakher, The Theory of Matrices (Nauka, Moscow, 1967; Chelsea, New York, 1959).
  4. A. V. Panteleev, A. S. Yakimova, and K. A. Rybakov, Ordinary Differential Equations. Practical Course (INFRA-M, Moscow, 2016) [in Russian].
  5. L. S. Pontryagin, Ordinary Differential Equations (Fizmatgiz, Moscow, 1961; Addison-Wesley, Reading, 1962).
  6. A. F. Filippov, Collection of Problems on Differential Equations (Fizmatgiz, Moscow, 1961) [in Russian].
  7. O. M. Muhamedjonova, “Jordan Matrix Form and Solutions of Linear Systems of Ordinary Differential Equations with Constant Coefficients,” Uchen. Zap. Khujand State Univ. Named after Academician B. Gafurov. Series: Natural and Economic Sci. No. 1, 20-26 (2017).
    https://www.elibrary.ru/item.asp?id=29217659 . Cited June 27, 2023.
  8. A. A. Baloev, “The Matrix-Algebraic Form of a Solution to the System of Linear Ordinary Differential Equations with Constant Coefficients,” Sib. Zh. Ind. Mat. 17 (3), 3-12 (2014).
    https://www.elibrary.ru/item.asp?id=21956533 . Cited June 27, 2023.
  9. I. N. Shchitov and E. N. Begun, “On One Method for Solving Systems of Linear Differential Equations with Constant Coefficients,” in Proc. Int. Conf. on Actual Problems of Radio and Film Technologies, St. Petersburg, Russia, November 16-17, 2021.
    https://www.elibrary.ru/item.asp?id=49540472 . Cited June 27, 2023.
  10. Yu. V. Malyshev and P. S. Atamanov, “About Solution of the System of Linear Differential Equations by Symbolic Methods,” Vestn. Chuvash Univ., No. 3, 155-159 (2011).
    https://www.elibrary.ru/item.asp?id=16771870 . Cited June 27, 2023.
  11. V. V. Ivlev and E. A. Krivoshey, “Systems of Linear Differential Equations. Integrable Combinations (Continued),” Math. Educ., No. 1, 47-51 (2018).
    https://www.elibrary.ru/item.asp?id=32834675 . Cited June 27, 2023.
  12. M. A. Rybakov, “Solving Systems of Linear Differential Equations with Constant Coefficients by Means of Laplace Transformation,” Vestn. Tambov State Univ. 14 (4), 791-792 (2009).
    https://www.elibrary.ru/item.asp?id=13035501 . Cited June 27, 2023.
  13. A. B. Nazimov and M. A. Ochilova, “On the Reduction of a System of Linear Differential Equations with Constant Coefficients to a Single High-Order Differential Equation,” in Modern Problems and Prospects for Teaching Mathematics, Physics, and Computer Science at School and University (Vologda State University, Vologda, 2021), pp. 41-47.
    https://www.elibrary.ru/item.asp?id=44908296 . Cited June 27, 2023.
  14. A. G. Podgaev and A. Z. Sin, “Simple Proof of Elimination Method under the Solution of Normal Linear System of Differential Equations with Constant Coefficients,” Uchen. Zap. Pacific National Univ. 5 (4), 1357-1363 (2014).
    https://www.elibrary.ru/item.asp?id=22674856 . Cited June 27, 2023.
  15. T. T. Ha and J. A. Gibson, “A Note on the Determinant of a Functional Confluent Vandermonde Matrix and Controllability,” Linear Algebra Appl. 30, 69-75 (1980).
    doi 10.1016/0024-3795(80)90182-2
  16. Yu. S. Antonov, “On the History of Solving Third and Fourth Degree Equations,” Sci. Technol. Yakutia, No. 2, 108-110 (2015).
    https://www.elibrary.ru/item.asp?id=37030535 . Cited June 27, 2023.
  17. M. M. Postnikov, Galois Theory (Fizmatgiz, Moscow, 1963) [in Russian].
  18. V. V. Karachik, “Method for Constructing Solutions of Linear Ordinary Differential Equations with Constant Coefficients,” Zh. Vychisl. Mat. Mat. Fiz. 52 (2), 237-252 (2012) [Comput. Math. Math. Phys. 52 (2), 219-234 (2012)].
    doi 10.1134/S0965542512020108

License

Copyright (c) 2023 Д. Н. Баротов, Р. Н. Баротов

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.