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which follows immediately when evaluation on x = ±1 is imposed. It is left as an
exercise (which may require a graphical approach) to confirm that (5.140) has zero,
one or two solutions for given depending on whether or
respectively, where the critical value is the solution of

it can be shown that there does exist just one It turns out that the
consequences of this are fundamental: for any if the initial temperature is high
enough, or for any temperature if then the time-dependent problem produces
a temperature that increases without bound—indeed, in a finite time. What
we will do here is to examine the temperature attained according to the steady-state
equation, (5.138), for various although we will approach this by considering different
sizes of temperature (as measured by
It is immediately apparent that the approximation that led to (5.139) cannot be valid
if the temperature is as large as see equation (5.138). Let us therefore write
and then (5.138) becomes

and so if we seek a solution (with as ) we obtain simply

where A and B are arbitrary constants. Such a solution is unable to accommodate
a maximum temperature at x = 0 (if the solution is to be differentiable, and
constant for all x does not satisfy (5.141)). Thus (5.142) can describe the solution
only away from x = 0, but then we may impose the boundary conditions on x = ±1,
so

and the single arbitary constant, A, may be used in both solutions by virtue of the
symmetry. Near x = 0, let and seek a solution where
is the (scaled) maximum temperature attained in the limit note
that, at this stage, we do not know the scalings and Equation (5.141) becomes