On the asymptotic behavior of solutions of third-order binomial differential equations

The paper discusses the development of a method for constructing asymptotic formulas for $x\to +\infty$ of a fundamental system of solutions of two-term singular symmetric differential equations of odd order with coefficients from a wide class of functions that allow oscillation (with weakened regularity conditions that do not satisfy the classical Titchmarsh–Levitan regularity conditions). Using the example of a third-order binomial equation $\left(i2\right)\left[\right(p{\left(x\right)y^{\text{'}}{)}^{\text{'}\text{'}}}^{}+\left(p\right(x\left){y^{\text{'}\text{'}}}^{}{)}^{\text{'}}\right]+q\left(x\right)y=\lambda y$ the asymptotics of solutions in the case of different behavior of the coefficients $qx$ is studied, $hx-1+1\sqrt{px}$. New asymptotic formulas are obtained for the case when $hx\notin L_1\infty$.