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
- 4 Downloads
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.
Key wordslinear equation differential operator eigenvalues multiperiodic solution
Unable to display preview. Download preview PDF.
- 1.Arnold, V.I. Ordinary Differential Equations (Springer-Verlag, Berlin-Heidelberg, 1992).Google Scholar
- 2.Arnold, V.I. Additional Chapters of Ordinary Differential Equations (Nauka. Moscow, 1978) [in Russian].Google Scholar
- 5.Kharasakal, V.Kh. Almost periodic solutions of ordinary differential equations (Nauka, Alma-Ata, 1970) [in Russian].Google Scholar
- 6.Kulzhumiyeva, A.A., Sartabanov, Zh.A. Periodic solutions of systems of differential equations with multidimensional time (RIT. ZKGU, Ural’sk, 2013) [in Russian].Google Scholar
- 7.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].Google Scholar
- 8.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].Google Scholar
- 11.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].Google Scholar
- 12.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].Google Scholar