In our discussion of critical damping (p = 600), the

QUESTION:

In our discussion of critical damping (p = 600), the second solution (5.43) was rather pulled out of a hat. One can arrive at it in a reasonably systematic way by looking at the solutions for /3 < too and carefully letting /3 coo, as follows: For /3 < wo, we can write the two solutions as x1(t) = e-Pt cos(wit) and x2(t) = e-13` sin(wit).' Show that as p wo, the first of these approaches the first solution for critical damping, x1(t) = e-Pt . Unfortunately, as /3 wo, the second of them goes to zero. (Check this.) However, as long as /3 wo, you can divide x2(t) by col and you will still have a perfectly good second solution. Show that as p wo, this new second solution approaches the advertised to-13`.