Chapter 5, Problem 72CE

In a study of heat transfer, we find that for a solid rod there is a relationship between the second derivative of the temperature with respect to position along the rod and the first with respect to time. (A linear temperature change with position would imply as much heat flowing into a region as out, so the temperature there would not change with time.)
$\frac{{\partial }^{2}T\left(x,t\right)}{\partial x^2}b\frac{\partial T\left(x,t\right)}{\partial t}$

1. Separate variables. That is, assume a solution that is a product of a function of x and a function of t, plug it in, then divide by it. Obtain two ordinary differential equations.
2. Consider a fairly simple, if somewhat unrealistic, case. Suppose the temperature is 0 at $x=0$ and $x=L$, and positive in between. Write down the simplest function of x that (1) fits these conditions and (2) obeys the differential equation involving x. Does your choice determine the value, including sing, of some constant?
3. Obtain the full $T\left(x,t\right)$ for this case.