This article discusses a linear system of inhomogeneous equations with a differentiation operator for a scalar argument. Cauchy problems for both homogeneous and inhomogeneous systems of equations are studied. The properties of helical characteristics and initial characteristic integrals are established. A new method for studying systems with two differentiation operators is developed based on the transition from a scalar argument to a vector argument. The analytical form of the unique solution to the initial problem for a system with two differentiation operators in a vector-matrix form is found. Integral representations of the unique solution to a system of equations with a differentiation operator in vector form, defined on a cylindrical surface, are obtained for the cases of trivial and periodic initial conditions. As a result of the study, a new method for solving initial value problems for linear homogeneous and inhomogeneous systems with two differentiation operators is developed, based on the transition to a system with one differentiation operator, from which a scheme for studying such systems is provided. The article uses the results and methods of the scientific project "Method of periodic characteristics in the study of oscillations in systems with a diagonal differentiation operator."

SARTABANOV ZH.A.

Doctor of Physical and Mathematical Sciences, Professor, Aktobe Regional University named after K. Zhubanov, Aktobe, Kazakhstan.

E-mail: sartabanov42@mail.ru, https://orcid.org/0000-0003-2601-2678

ZHUMAGAZIEV A.KH.

PhD, Aktobe Regional University named after K. Zhubanov, Aktobe, Kazakhstan.

E-mail: azhumagaziyev@zhubanov.edu.kz, https://orcid.org/0000-0002-6007-3311

KHAMIMOVA Z.K.

Master's student, Aktobe Regional University named after K. Zhubanov, Aktobe, Kazakhstan.

E-mail: hamimova.zarina@mail.ru, https://orcid.org/0009-0007-3871-3275

1. Zhumagaziyev A.Kh., Sartabanov Zh.A., Sultanaev Ya.T. On a new method for investigation of multiperiodic solutions of quasilinear strictly hyperbolic system // Azerbaijan Journal of Mathematics, (2022), Vol. 12, №1, P. 32-48. (WoS Q2) WOS:000824351800003.
2. Umbetzhanov D.U., Pochti periodicheskie resheniya evolyucionnyh uravnenij, Alma-Ata: Nauka, (1990), 184 s.
3. Sartabanov ZH.A. Periodichnost' harakteristik operatora differencirovaniya po diagonali// Vestnik KazNPU im. Abaya, seriya «Fiziko-matematicheskie nauki». -2023, - T.82. - №2. - S. 40-53. 2 vmesto 3
4. Harasahal V.H., Pochtiperiodicheskie resheniya obyknovennyh differencial'nyh uravnenij, Alma-Ata: Nauka, (1970),-200
5. Gantmaher F.R. Teoriya matric. – M.: FIZMATLIT, 2010. – 560 s.
6. Matrosov, V.L. Differencial'nye uravneniya i uravneniya s chastnymi proizvodnymi: Uchebnik dlya studentov vuzov / V.L. Matrosov, R.M. ogly Aslanov, M.V. Topunov. - M.: VLADOS, 2011. - 376 c.

differentiation operator, initial conditions, uniqueness of the solution, periodicity, helical characteristics, linear system, matrix