4 7.28, §4 Solve the Other ODE?

If you look back to the beginning of the previous section, you may wonder about the function $T(t)$. It satisfied

$$\frac{T''}{a^2 T} = -\lambda$$

Now that we have found the values for $\lambda$ from the $X$-problem, we could solve this ODE too, and find functions $T_1(t), T_2(t),\ldots$. Many people do exactly that. However, if you want to follow the crowd, please keep in mind the following:

1. The values of $\lambda$ can only be found from the Sturm-Liouville problem for $X$. The problem for $T$ is not a Sturm-Liouville problem and can never produce the correct values for $\lambda$.
2. The functions $T(t)$ do not satisfy the same initial conditions at time $t=0$ as $u$ does.
3. Finding $T$ is useless if the PDE is inhomogeneous; it simply does not work. (Unless you add still more artificial tricks to the mix, as the book does.)

We will just ignore the entire $T$. Instead in the next section we will systematically solve the problem for $u$ without tricks using our found eigenfunctions. What we do there will always work. If you want to try to take a shortcut for an homogeneous PDE, well, the responsibility and risk are yours alone. Someday I will stop seeing students getting themselves in major trouble this way at the final, but it may not be this year.