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4 1. Mathematical preliminaries

procedure has generated a solution which is not periodic and for which the amplitude
grows without bound as Yet the exact solution is simply

which is easily obtained by scaling out the factor, by working with
rather than t. (The ‘e’ subscript here is used to denote the exact solution.) It is now an
elementary exercise to check that (1.8) and (1.7) agree, in the sense that the expansion
of (1.8), for small and fixed t, reproduces (1.7). (A few examples of expansions
are set as exercises in Q1.1, 1.2.) This process immediately highlights one of our
difficulties,namely, taking first and then allowing this is a classic case
of a non-uniform limiting process i.e. the answer depends on the order in which the limits
are taken. (Examples of simple limiting processes can be found in Q1.4.) Clearly, any
approximate methods that we develop must be able to cope with this type of behaviour.
So, for example, if it is known (or expected) that bounded, periodic solutions exist,
the approach that we adopt must produce a suitable approximation to this solution.
We have taken some care in our description of this first example because, at this
stage, the approach and ideas are new; we will present the other examples with slightly
less detail. However, before we leave this problem, there is one further observation
to make. The original equation, (1.1), can be solved easily and directly; an associated
problem might be

with appropriate initial data. This describes an oscillator for which the frequency
depends on the value of x(t) at that instant—it is a nonlinear problem. Such equations
are much more difficult to solve; our techniques have got to be able to make some
useful headway with equations like (1.9).

E1.2 A first-order equation
We consider the equation

with Again, let us seek a solution in the form

and then obtain

or