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4. THE METHOD OF MULTIPLE SCALES

The final stage in our presentation of the essential tools that constitute singular per-
turbation theory is to provide a description of the method of multiple scales, arguably the
most important and powerful technique at our disposal. The idea, as the title implies,
is to introduce a number of different scales, each one (measured in terms of the small
parameter) associated with some property of the solution. For example, one scale might
be that which governs an underlying oscillation and another the scale on which the
amplitude evolves (as in amplitude modulation). Indeed, this type of problem is the most
natural one with which to start; we will explore a particularly simple example and
use this as a vehicle to present the salient features of the method. However, before we
embark on this, one word of warning: this process necessarily transforms all differen-
tial equations into partial differential equations—even ordinary differential equations!
This could well cause some anxiety, but the comforting news is that the underlying
mathematical problem is no more difficult to solve. So, for example, an ordinary dif-
ferential equation, subjected to this procedure, involves an integration method that
is essentially unaltered; the only adjustment is simply that arbitrary constants become
arbitrary functions of all the other variables.

4.1 NEARLY LINEAR OSCILLATIONS
We will show how these ideas emerge in this class of relatively simple problems; indeed,
we start with an example for which an exact solution exists. Let us consider the linear,