A cylindrical metallic rod of length $L$ and radius $R$ is perfectly insulated along its curved surface. One end is maintained at temperature $T_1$ and the other at $T_2$, where $T_1 > T_2$. The material's thermal conductivity varies linearly with temperature, given by $k(T) = k_0(1 + \alpha T)$, where $k_0$ and $\alpha$ are positive constants.