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In this paper, we model, classify and investigate the solutions of(normalized) second order ode's with \emph{nonconstant continuouscoefficients}. We introduce a generalized \emph{frequency function} as thesolution of a \emph{nonlinear integro-differential equation}, show itsexistence and then derive a representation formula for (all) solutions of(normalized) second order ode's with \emph{nonconstant continuouscoefficients}. Because this formula specifies the interplay between thecoefficients of the ode, the \emph{relaxation function} ("strongly" decreasingpositive function) and the frequency function of the oscillation, it can beapplied to design models of dissipative oscillations. As an application, wepresent and discuss some oscillation models that stop within a finite timeperiod. Moreover, we demonstrate that a large class of oscillations can be usedto design and analyze dissipative waves. In particular, it is easy to modeldissipative waves that cause in each point of space an oscillation that stopsafter a finite time period.

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