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

Yaudat T. Sultanaev, Nur F. Valeev and Elvira A. Nazirova ()
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Yaudat T. Sultanaev: Faculty of Physics and Mathematics, Bashkir State Pedagogical University n. a. M. Akmulla, Ufa 450008, Russia
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
Elvira A. Nazirova: Institute of Informatics, Mathematics and Robotics, Ufa University of Science and Technology, Ufa 450074, Russia

Mathematics, 2024, vol. 12, issue 2, 1-11

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.

Keywords: asymptotic methods; oscillating coefficients; singular differential equations of odd order; Campbell’s identity; quasi-derivatives; Shin–Zettl matrix (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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