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8 1. Mathematical preliminaries

not satisfy and the solution

does not satisfy Indeed, we have no way of knowing which, if either, is
correct; thus there is little to be gained by solving the problem:

(We note that, since we must have and then there is, ex-
ceptionally, a solution of the complete problem: for But we still
do not know
As in our previous examples, let us construct and examine the exact solution. Equa-
tion (1.16) is a second order, constant coefficient, ordinary differential equation and
so we may seek a solution in the form

i.e.

The general solution is therefore

and, imposing the two boundary conditions, this becomes

(We can note here that the contribution from the term is absent in the
special case we proceed with the problem for which
This solution, (1.22), is defined for and with let us select any
and, for this x fixed, allow (where denotes tending to zero
through the positive numbers). We observe that the terms and
vanish rapidly in this limit, leaving

this is our approximate solution given in (1.20). (Some examples that explore the
relative sizes of exp(x) and ln(x) can be found in Q1.5.) Thus one of the possible
options for introduced above, is indeed correct. However, this solution is, as
already noted, incorrect on x = 0 (although, of course, The difficulty
is plainly with the term for any x > 0 fixed, as this vanishes
exponentially, but on x = 0 this takes the value 1 (one). In order to examine the rôle
of this term, as we need to retain it (but not to restrict ourselves to x = 0); as