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38 1. Mathematical preliminaries

The error in using this composite expansion is, with

But the maximum value of occurs at for then

and so we have the estimate

and hence the error, for is at most (indeed, this upper bound is approx-
imately for small

This example demonstrates that estimates for the error in using a composite expansion
are readily derived, at least in particular cases, when the original function is known.
However, corresponding results for the solutions of differential equations are not so
easily obtained—and this is the situation of most practical interest. Then we have
available only the equation and its asymptotic solution, but not the exact solution of
the equation, of course. Thus any estimates will need to be based on an analysis of
appropriate differential inequalities; we will touch on these ideas later in this text. (We
should add that only rarely is it possible satisfactorily to complete such calculations; in
addition, it is often deemed not worthwhile to devote much energy to this exercise,
essentially because the results and the context persuade us that we have the appropriate
and correct asymptotic form of the solution.)