Oscillatory criteria for the second order linear ordinary differential equations in the marginal sub extremal and extremal cases

TL;DR

This paper develops new oscillatory criteria for second order linear differential equations using Riccati equations, including an extremal condition for Mathieu's equation, enhancing understanding of oscillation behavior in various cases.

Contribution

It introduces three novel oscillatory criteria for second order linear ODEs, extending and comparing with existing criteria, and provides an extremal condition for Mathieu's equation.

Findings

New oscillatory criteria established for different cases.

The first criterion generalizes J. Deng's criterion.

An extremal oscillatory condition for Mathieu's equation is derived.

Abstract

The Riccati equation method is used to establish three new oscillatory criteria for the second order linear ordinary differential equations in the marginal, sub extremal and extremal cases.We show that the first of these criteria implies the J. Deng's oscillatory criterion. An extremal oscillatory condition for the Mathieu's equation is obtained. The obtained results are compared with some known oscillatory criteria.