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72 2. Introductory applications

E2.9 An equation with an interesting behaviour near x = 0
We consider the problem

as we assume that

for some x = O(1). We obtain the sequence of equations

and so on; the boundary condition (if available here) gives

The general solution of (2.42a) is simply

(which is not defined on x = 0, so we anticipate the need for a scaling as if a
bounded solution exists), and then (2.42b) gives

or

The asymptotic expansion is therefore

and this is defined for x = O(1), including x = 1, but not as So the boundary
condition, (2.43), can be applied, requiring the arbitrary constants to be and
i.e.