= Solution
{c}
Assume the rod has constant <cross-sectional area> $A$, no internal source of <heat>, no bulk motion, and no lateral <heat flux>. Assume also that each section has a uniform <temperature>, so that $T$ depends only on $x,t$, and that <Fourier's law> applies with the stated constant <thermal conductivity> $k$. The longitudinal <heat flux density> is then $q=-kT_x$.
By <conservation of energy>, the rate of increase of <internal energy> in a slice $[x,x+\Delta x]$ equals incoming minus outgoing <heat>:
$$
\rho c A\int_x^{x+\Delta x}T_t(s,t)\,ds
=A\bigl(q(x,t)-q(x+\Delta x,t)\bigr).
$$
Divide by $A\Delta x$ and take the <limit> $\Delta x\to0$. For a sufficiently differentiable <temperature> field this gives $\rho cT_t=-q_x=kT_{xx}$, and hence
$$
\boxed{T_t=\kappa T_{xx},\qquad\kappa=\frac{k}{\rho c},\qquad0<x<L,\quad t>0.}
$$
Here $\rho$ is <mass density>, $c$ is <specific heat capacity>, and $\kappa$ is <thermal diffusivity>.
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