The heat equation and a nonlinear counterpart predict the temperature at the end of a semi-infinite rod, when the heat flux is specified there. Both equations have reductions to ordinary differential equations which simplify their solutions. For the nonlinear equation, the problem becomes
y
(
z
)
=
y
(
z
)
2
−
z
y
(
z
)
y
′
(
z
)
,
y \left ( z\right ) =y\left ( z\right ) ^{2}-zy\left ( z\right ) y^{\prime }(z),
y
′
(
0
)
=
−
3
,
y
(
∞
)
=
0
\ y^{\prime }\left ( 0\right ) =-\sqrt {3},y\left ( \infty \right ) =0
on
[
0
,
∞
)
[0,\infty )
. A variational approximation to the initial value
y
(
0
)
y\left ( 0\right )
of the nonlinear boundary value problem was found by employing an integral constraint suggested by the equation itself. The approximation is more accurate for the initial value than that of Mickens and Wilkins but misses their pointwise description of the solution.