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

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