PARADIGMS OF ORDINARY DIFFERENTIAL-EQUATIONS

Of the many ordinary differential equations which arise in modelling v arious aspects of the Universe, a goodly proportion are variants of a few standard equations. If the equation is lineal; the paradigm is the time-dependent harmonic oscillator Its treatment leads naturally to t he Pinney equation and its generalizations to higher order and higher dimension. In the latter it is intimately associated with Ermakov syst ems and the enigmatic magnetic monopole. The nonlinear paradigm is the Emden-Fowler equation, which had its genesis in theoretical astrophys ics. Its analysis leads to Mahomed's equation, which is of interest in the study of nonlinear equations from both the Lie and Painleve appro aches. The final paradigm is the first order Abel's equation, which is usually the graveyard of hopes for practical integrability.