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Equation (4.15) can be written

where is an arbitrary constant; but (4.16) then requires

Equation (4.14) can be written in a similar fashion:

and it follows that the solution for will contain a term unless
We must make this choice in order to avoid a non-uniformity as (because this
second term in the asymptotic expansion will grow like relative to the first). Thus
we are left with

and hence, noting (4.17), we have (and the arbitrary is now unim-
portant). Thus we have the solution

where this should be compared with the expansion of the exact
solution, (4.2).

We have used this example to introduce and illustrate all the essential features of the
technique. It should be clear that the transformation from an ordinary to a partial
differential equation does not introduce any undue complications in the method of
solution. We do see that we must impose periodicity and uniformity at each order, and
that this produces conditions that uniquely describe the solution at the previous order.
Indeed, the removal of terms that generate secularities is fundamental to the approach.
Further, as we have seen, suitable freedom in the choice of the fast scale enables non-
uniformities also to be removed. The only remaining question, at least in the context
of ordinary differential equations, is how the method fares when the equation cannot
be solved exactly (so we have no simple guide to the form of solution, as we did above).
We explore this aspect via another example.

E4.2 A Duffing equation with damping
We consider the problem

with