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60 2. Introductory applications

written for this purpose (with a prime for the derivative) as

gives

and so on, so that takes the form (for appropriate functions
When this solution is used to generate it is clear that we will produce terms
in and and so this pattern will continue: the equation implies the ‘natural’
asymptotic sequence so this is what we will assume to initiate the solution method.
(It should be noted that the boundary condition is consistent with this assumption,
as is the alternative condition On the other hand, a boundary value
would force the asymptotic sequence to be adjusted to accommodate
this i.e.
Thus we seek a solution of the problem (2.11) in the form

for some (and we do not know which xs will be allowed, at this stage). The
expansion (2.12) is used in the differential equation to give

where ‘= 0’ means zero to all orders in thus we require

and so on. Similarly, the boundary condition gives

so that

of course, to evaluate on x = 1 implies that the asymptotic expansion, (2.12), is valid
here—but we do not know this yet. This is written down because, if the problem turns
out to be well-behaved i.e. regular, then we will have this ready for use; essentially,
all we are doing is noting (2.14)—we can reject it if the expansion will not permit
evaluation on x = 1.
The next step is simply to solve each equation (for in turn; we see directly
(from (2.13a)) that the general solution for is