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

John A. D. Appleby, Subham Pal

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.

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

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.

Paper Structure

This paper contains 7 sections, 15 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Representation of Solutions; Characterisation of Convergence of $x$ and $x'$
  3. Representations of $x$, $y_2$ and $F$
  4. Representation of $x_0$ and $y$ in terms of $F$
  5. Characterisation of Convergence, Boundedness and Unboundedness
  6. Uniform Convergence of $F_\Theta$
  7. Stability with Unbounded and Highly Oscillatory Forcing

Key Result

Theorem 1

Let $f\in L^1_{loc}([0,\infty);\mathbb{R})$ and let $x$ be the unique continuous solution of the perturbed differential equation eq.x. Suppose moreover that $a>0$, $b>0$. Let $F_{(\theta_1,\theta_2)}$ be defined by eq.Ftheta1theta2 and $y_2$ by eq.y2. Then the following are equivalent:

Theorems & Definitions (24)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • proof
  • Theorem 7
  • proof
  • Lemma 8
  • ...and 14 more