Abstract
We consider an n-th order linear equation with differential operator corresponding to the direction of the main diagonal in the space of independent variables; its coefficients. in general, are variable but constant on the diagonal. We establish conditions on variable eigenvalues which give a possibility to apply some known methods of the theory of for ODEs, when integrating the equation. On this basis, the structures of solutions to the homogeneous equation are determined. We give conditions for existence of multiperiodic solutions of the equations related to variable eigenvalues and initial functions. An integral representation of multiperiodic solution to the inhomogeneous equation is given. We also suggest the concepts of variable frequency and variable period.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arnold, V.I. Ordinary Differential Equations (Springer-Verlag, Berlin-Heidelberg, 1992).
Arnold, V.I. Additional Chapters of Ordinary Differential Equations (Nauka. Moscow, 1978) [in Russian].
Fedoryuk, M.V. Ordinary Differential Equations (Nauka. Moscow, 1980) [in Russian]
Umbetzhanov, D.U. Almost multiperiodic solutions of partial differential equations (Nauka, Alma-Ata, 1979) [in Russian]. 1
Kharasakal, V.Kh. Almost periodic solutions of ordinary differential equations (Nauka, Alma-Ata, 1970) [in Russian].
Kulzhumiyeva, A.A., Sartabanov, Zh.A. Periodic solutions of systems of differential equations with multidimensional time (RIT. ZKGU, Ural’sk, 2013) [in Russian].
Kulzhumiyeva, A.A. “Investigation of periodic solutions o. systems, reduced to canoni. form, with linear operator by multidimensional time”, Evraziiskii Matem. Zh. 2, 69–73 (2008) [in Russian].
Kulzhumiyeva, A.A., Sartabanov, Zh.A. “On reducibility of linear D e-system with constant coefficients on the diagonal and Jordan matrix in the case of its equivalence to equation of higher order”, Vestn. KarGU, Ser. Matem. 4, 88–94 (2016) [in Russian].
Kulzhumiyeva, A.A., Sartabanov, Zh.A. “Reduction of linear homogeneous D e-systems to the Jordan canonical form”, Izv. NAN Resp. Kazakhstan, Ser. Phys.-Mat. 315(5), 5–12 (2017).
Kulzhumiyeva, A.A., Sartabanov, Zh.A. “On multiperiodic integrals of a linear system with the differentiation operator in the direction of the main diagonal in the space of independent variables”, Eurasian Math. J. 8(1), 67–75 (2017).
Kulzhumiyeva, A.A., Sartabanov, Zh.A. “Constructing of periodic solution of a quasilinear system”, Vestn. ENU im. L.N. Gumileva 121(6), 25–29 (2017) [in Russian].
Sartabanov, Zh.A., Kulzhumiyeva, A.A. “Reducibility of linear multiperiodic equations with differential operator on the diagonal”, Matem. Zhurn. 67(1), 139–150 (2018) (Inst. matem. i matem. model., Almati) [in Russian].
Author information
Authors and Affiliations
Corresponding authors
Additional information
Russian Text © The Author(s), 2019, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2019, No. 6, pp. 34–47.
About this article
Cite this article
Kulzhumiyeva, A.A., Sartabanov, Z.A. Integration of a Linear Equation with Differential Operator, Corresponding to the Main Diagonal in the Space of Independent Variables, and Coefficients, Constant on the Diagonal. Russ Math. 63, 29–41 (2019). https://doi.org/10.3103/S1066369X19060045
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066369X19060045