In Section 2.10, this problem was solved as an example:

   

The solution was found in the form of a Fourier integral,

   

Here are animations of the solution with parameters b = 1 and k = 1 for the time t = 0 to 2.
On the left is an animation with equally spaced times from t = 0 to 2.
On the right, the animation uses logarithmically spaced times. This makes it easier to see how the solution changes at times near 0.