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

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 \(h(x) \notin L_1[1,\infty ) \).