On the Asymptotic Behavior of Solutions of Third-Order Binomial Differential Equations

Pub Date : 2024-06-05 DOI:10.1134/s0012266124020095

Ya. T. Sultanaev, N. F. Valeev, E. A. Nazirova

引用次数: 0

Abstract

The paper discusses the development of a method for constructing asymptotic formulas as \(x\to \infty \) for the fundamental solution system of two-term singular symmetric differential equations of odd order with coefficients in a broad class of functions that allow oscillation (with relaxed regularity conditions that do not satisfy the classical Titchmarsh–Levitan regularity conditions). Using the example of a third-order binomial equation \(({i}/{2})\bigl [(p(x)y^{\prime })^{\prime \prime }+(p(x)y^{\prime \prime })^{\prime }\bigr ] +q(x)y =\lambda y\), the asymptotics of solutions in the case of various behavior of the coefficients \(q(x)\) and \(h(x)=-1+{1}\big /{\sqrt {p(x)}}\) is studied. New asymptotic formulas are obtained for the case in which \(h(x) \notin L_1[1,\infty ) \).

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论三阶二项式微分方程解的渐近行为

Abstract The paper discusses the development of a method for constructing asymptotic formulas as\(x\to \infty \) for the fundamental solution system of two-termsingular symmetric differential equation of odd order with coefficients in a wide class offunctions that allow oscillation (with relaxed regularity conditions that not satisfy the classicalTitchmarsh-Levitan regularity conditions).以三阶二项式方程为例（({i}/{2})\bigl [(p(x)y^{\prime })^{\prime }+(p(x)y^{\prime })^{\prime }\bigr ] +q(x)y =\lambda y\ ）、和（h(x)=-1+{1}\big /\sqrt {p(x)}\）的各种行为情况下的解的渐近性进行了研究。对于 \(h(x) \notin L_1[1,\infty ) 的情况，得到了新的渐近公式。\).

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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