Advanced Engineering Mathematics, 7th Edition

11.4 Autonomous Systems as Mathematical Models

INTRODUCTION

Many applications from physics give rise to nonlinear autonomous second-order differential equations—that is, DEs of the form x″ = g(x, x′). For example, in the analysis of free, damped motion of a spring/mass system in Section 3.8 we assumed that the damping force was proportional to the velocity x′ and the resulting model mx″ = –βx′ – kx is a linear differential equation. But if the magnitude of the damping force is proportional to the square of the velocity, the new differential equation mx″ = –βx′ – kx is nonlinear. The corresponding plane autonomous system is nonlinear:

In this ...