# A representation formula for solutions of second order ode's with time dependent coefficients and its application to model dissipative oscillations and waves

## TL;DR

This paper introduces a new representation formula for solutions of second order ODEs with time-dependent coefficients, enabling modeling of dissipative oscillations and waves that cease within finite time.

## Contribution

It develops a generalized frequency function via a nonlinear integro-differential equation and derives a solution representation formula for second order ODEs with nonconstant coefficients.

## Key findings

- The representation formula clarifies the relationship between coefficients, relaxation, and frequency functions.
- It allows designing models of dissipative oscillations that stop in finite time.
- The approach extends to modeling dissipative waves with localized oscillations.

## Abstract

In this paper, we model, classify and investigate the solutions of (normalized) second order ode's with \emph{nonconstant continuous coefficients}. We introduce a generalized \emph{frequency function} as the solution of a \emph{nonlinear integro-differential equation}, show its existence and then derive a representation formula for (all) solutions of (normalized) second order ode's with \emph{nonconstant continuous coefficients}. Because this formula specifies the interplay between the coefficients of the ode, the \emph{relaxation function} ("strongly" decreasing positive function) and the frequency function of the oscillation, it can be applied to design models of dissipative oscillations. As an application, we present and discuss some oscillation models that stop within a finite time period. Moreover, we demonstrate that a large class of oscillations can be used to design and analyze dissipative waves. In particular, it is easy to model dissipative waves that cause in each point of space an oscillation that stops after a finite time period.