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68 2. Introductory applications

but the boundary condition is not available, because this is specified where x = O(1).
Equation (2.28) suggests that we seek a solution in the form

which gives

Immediately we obtain (an arbitrary constant), and then equation (2.29b)
becomes

which integrates to give

where is a second arbitrary constant.
The resulting 2-term expansion is therefore

the two arbitrary constants are determined by invoking the matching principle: (2.30)
and (2.27) are to match. Thus we write the terms in (2.27) as functions of X, let
(for X = O(1)) and retain terms O(1) and (which are used in (2.30)); conversely,
write (2.30) as a function of x, expand and retain terms O(1), and From
(2.27) we construct

and from (2.30) we write

(This expansion requires the standard result: