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The 2-term asymptotic expansion, is uniformly valid for
Let us find the next term in the expansion; this is the solution of

Thus, introducing the integrating factor we have

and so

where the arbitrary constant must be A = 0 (to satisfy This third term
in the asymptotic expansion is very different from the first two: it is not defined on
x = 0, so we must expect a breakdown. The expansion, to this order, is now

as for x = O(1); as we clearly have a breakdown where the second
and third terms in the expansion become the same size i.e. or
Note that this breakdown occurs for a larger size of x (as x is decreased from O(1))
than the breakdown associated with the first and third terms, so we must consider
The problem for is formulated by writing

where the relabelling of y is an obvious convenience (and we note that y = O(1) for
The original equation, in (2.26), expressed in terms of X and Y, requires
the identity

and then we obtain