Chapter 12.4, Problem 87E

Heat equation The flow of hear along a thin conducting bar is governed by the one-dimensional heal equation (with analogs for thin plates in two dimensions and for solids in three dimensions)

     $\frac{\partial u}{\partial t}=k\frac{{\partial }^{2}u}{\partial x^2},$

where u is a measure of the temperature at a location x on the bar at time t and the positive constant k is related to the conductivity of the material. Show that the following functions satisfy the heal equation with k = 1.

87.     $u\left(x,t\right)=A{e}^{-a^{2}t}$ cos ax, for any real numbers a and A