Characterisation of Convergence, Boundedness and Unboundedness in Solutions of Second Order Linear Differential Equations

TL;DR

This paper characterizes the conditions under which solutions of forced second order linear differential equations converge, remain bounded, or become unbounded, highlighting the effects of forcing terms and oscillations on solution behavior.

Contribution

It provides a new characterization of solution behaviors for forced second order linear differential equations, including convergence, boundedness, and unboundedness.

Findings

Solutions can tend to zero despite unbounded forcing terms.

Bounded solutions can exhibit high-frequency oscillations.

Unbounded derivatives do not necessarily imply unbounded solutions.

Abstract

This paper develops a characterisation of when solutions of forced second order linear differential equations converge to the zero solution of the asymptotically stable and unforced second order equation, or when the solution is bounded, but not convergent, or is unbounded. We see thereby that forcing terms can exhibit unbounded and high--frequency oscillation, and yet the solution may still tend to zero, even though the first and second derivative may become unbounded.