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260 5. Some worked examples arising from physical problems

and for an appropriate solution to exist, to leading order for X = O(1), we must have

which implies that is exponentially small as Let us write
then from (5.144) with (5.145) included, we obtain the equation for as

and this gives a meaningful first approximation, independent of only if we choose
e.g. This equation then has the general solution

for arbitrary constants and C. This solution is to be symmetric about X = 0, so
(which is satisfied with ), and is to match to
(5.143). Thus we must have and then we obtain

which matches only if, first, we choose the scaling and then
(valid in X > 0, X < 0, respectively). Thus, in particular, we find that

and so the maximum temperature becomes

where, from (5.145), we have

for given we may write this equivalently as where
and so determines is the interpretation that we employ.
The calculation thus far indicates that, for suitable the resulting steady-state
temperature (if it can be attained through a time-dependent evolution) is already very
large, namely But it is also clear from (5.147) that even smaller exist that give