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On the Asymptotic of Solutions of Odd-Order Two-Term Differential Equations

Author

Listed:
  • Yaudat T. Sultanaev

    (Faculty of Physics and Mathematics, Bashkir State Pedagogical University n. a. M. Akmulla, Ufa 450008, Russia
    Center for Applied and Fundamental Mathematics of Moscow State University, Moscow 119991, Russia
    These authors contributed equally to this work.)

  • Nur F. Valeev

    (Institute of Mathematics with Computing Centre—Subdivision of the Ufa Federal Research Centre of the Russian Academy of Sciences, Ufa 450008, Russia
    These authors contributed equally to this work.)

  • Elvira A. Nazirova

    (Institute of Informatics, Mathematics and Robotics, Ufa University of Science and Technology, Ufa 450074, Russia
    These authors contributed equally to this work.)

Abstract

This work is devoted to the development of methods for constructing asymptotic formulas as x → ∞ of a fundamental system of solutions of linear differential equations generated by a symmetric two-term differential expression of odd order. The coefficients of the differential expression belong to classes of functions that allow oscillation (for example, those that do not satisfy the classical Titchmarsh–Levitan regularity conditions). As a model equation, the fifth-order equation i 2 p ( x ) y ‴ ″ + p ( x ) y ″ ‴ + q ( x ) y = λ y , along with various behaviors of coefficients p ( x ) , q ( x ) , is investigated. New asymptotic formulas are obtained for the case when the function h ( x ) = − 1 + p − 1 / 2 ( x ) ∉ L 1 [ 1 , ∞ ) significantly influences the asymptotics of solutions to the equation. The case when the equation contains a nontrivial bifurcation parameter is studied.

Suggested Citation

  • Yaudat T. Sultanaev & Nur F. Valeev & Elvira A. Nazirova, 2024. "On the Asymptotic of Solutions of Odd-Order Two-Term Differential Equations," Mathematics, MDPI, vol. 12(2), pages 1-11, January.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:2:p:213-:d:1315468
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    References listed on IDEAS

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    1. Rossmann, Wulf, 2006. "Lie Groups: An Introduction Through Linear Groups," OUP Catalogue, Oxford University Press, number 9780199202515.
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