Separation of Variables and the Heat Equation

The heat equation $u_t = k\,u_{xx}$ models temperature $u(x,t)$ diffusing along a rod ($k > 0$ is the thermal diffusivity). Separation of variables looks for solutions of the special product form

Substituting and dividing by $k\,X\,T$ gives

The left side depends only on $t$, the right only on $x$; the only way they can be equal for all $x, t$ is if both equal the same constant $-\lambda$. This splits one PDE into two ordinary ones.

Worked example. From the split, $T' = -\lambda k\,T$ gives a decaying time factor $T(t) = e^{-\lambda k t}$, while $X'' = -\lambda X$ gives oscillating $X(x) = \sin(\sqrt\lambda\, x)$ (with suitable boundary conditions). So each mode looks like $u = e^{-\lambda k t}\sin(\sqrt\lambda\, x)$. For $k = 4$ and spatial part $\sin(2x)$ (so $\sqrt\lambda = 2$, $\lambda = 4$), the decay rate is $\lambda k = 4\times 4 = 16$: $u = e^{-16t}\sin(2x)$. As $t \to \infty$, every such mode decays to $0$.