The behavior of solutions of the systems of two first order linear ordinary differential equations

TL;DR

This paper investigates the behavior of solutions to systems of two first-order linear ordinary differential equations using the Riccati equation method, revealing oscillation types, stability criteria, and generalizing Leighton's theorem.

Contribution

It introduces new principles for second-order linear differential equations and provides stability and non-conjugation criteria for these systems.

Findings

All types of oscillation and regularity are characterized.

A generalization of Leighton's theorem is established.

Stability and non-conjugation criteria are proved.

Abstract

The Riccati equation method is used for study the behavior of solutions of the systems of two linear first order ordinary differential equations. All types of oscillation and regularity of these system are revealed. A generalization of Leighton's theorem is obtained. Three new principles for the second order linear differential equations are derived. Stability and non conjugation criteria are proved for the mentioned systems, as well as estimates are obtained for the solutions of the last ones.