A program for predicting the intervals of oscillations in the solutions of ordinary second-order linear homogeneous differential equations

James Graham-Eagle,Dimitris M Christodoulou,Qutaibeh D Katatbeh

doi:10.1186/s13662-016-0774-x

Abstract

We derive a new criterion for deducing the intervals of oscillatory behavior in the solutions of ordinary second-order linear homogeneous differential equations from their coefficients. The validity of the method depends on one’s ability to transform a given differential equation to its simplest possible form, so a program must be executed that involves transformations of both variables before the criterion can be applied. The payoff of the program is the detection of oscillations precisely where they may occur in finite or infinite intervals of the independent variable. We demonstrate how the oscillation-detection program can be carried out for a variety of well-known differential equations from applied mathematics and mathematical physics.

Highlights

1 Introduction In this work, we set out to investigate an apparently simple question concerning the theory of oscillatory solutions of ordinary second-order linear homogeneous differential equations: It is well known that such equations in the form y + By + Cy =, ( )
We provide a related example in Section . , where we study the Riemann-Weber [ ] equations, a long-standing counterexample to the discovery of a robust criterion for oscillatory solutions by considering the q(x) term alone of a differential equation given in the canonical form ( )
2.4 Constant damping and the criterion for oscillatory solutions The differential equations that we study can all be cast in the canonical form ( ), and this form can always be recast in the form ( ) with constant damping, but in the most commonly occurring cases, the coefficient of u(t) will not be constant

Summary

Introduction

This is the reason that in the past the canonical form has been proven inadequate in predicting the oscillatory behavior of the solutions of equation ( ) since the transformation to this form does not fold a pure-damping term into the coefficient q(x) of equation ( ) Where B and C are constants, shows that a criterion for oscillatory solutions can be established when a given differential equation can be transformed to a form that contains constant coefficients.

Results

Conclusion