Using the separation of variables method, write all the calculations needed to find
the the solution $T(t, x)$ of the heat equation
$\partial_t T = 5 \partial_x^2 T$, $t \in (0, \infty)$, $x \in (0, 1)$,
with boundary conditions
$T(t, 0) = 0$, $\partial_x T(t, 1) = 0$,
and initial condition
$T(0, x) = \tau(x) = \begin{cases} 1 & x \in [0, \frac{1}{2}] \\ 0 & x \in (\frac{1}{2}, 1] \end{cases}$
Note: The solutions of a Sturm-Liouville problem can be written without the need of
a proof.
