Introductory Remarks. The present Chapter deals with motions governed by linear ordinary differential equations with constant coefficients. The language of the dynamics of material systems will be used throughout, but the treatment can, for instance, be applied equally well to electrical systems. The discussion and exemplification of approximate numerical methods of solution is reserved for Chapter x.

Disturbed Motions. Except with very special systems or types of motion the differential equations which arise in dynamics are non-linear and do not admit exact solution. It is, however, sometimes possible to obtain particular solutions, such as those corresponding to equilibrium or steady motion. Then, if the system is supposed to be slightly disturbed from this known condition, the resulting small motion of deviation will be given by a set of linear differential equations. In the special case where the undisturbed state of the system is one of equilibrium or steady motion, the equations of disturbed motion will have constant coefficients and will be soluble by the methods of Chapters v and vi. In more general cases the equations will have for coefficients given functions of time, and they will thus be of the types considered in Chapter vii.

The disturbances just referred to may be of two kinds. They may be merely temporary, and represented by a set of initial conditions of motion which differ slightly from those corresponding to the undisturbed motion.