On the Oscillations of Second Order Linear Differential Equations

TL;DR

This paper generalizes the concept of the discriminant for second order linear differential equations, showing how its bounded behavior determines whether solutions oscillate infinitely often, making the criterion accessible to undergraduates.

Contribution

It extends the discriminant concept to general second order equations and links its boundedness to oscillatory behavior of solutions.

Findings

Discriminant bounds determine oscillation of solutions.

Provides an accessible criterion for oscillation analysis.

Connects classical discriminant theory with solution behavior.

Abstract

This paper extends the discriminant associated to second order linear constant coefficient differential equations to general second order linear differential equations. The main result of this paper is that the discriminant of a second order linear differential equation is a function who bounded behaviour determines whether solutions oscillate on an infinite interval, i.e. has infinitely many zeroes. This paper is accessible to any undergraduate who has completed a course in differential equations, and basic analysis.