Classical Mechanics - Special Interest Problem

For the underdamped case of the harmonic oscillator with viscous damping, the general solution derived in class was

x(t) = e^-bt ( A₁e^iwt + A₂e^-iwt )

Using the identity e^iq = cosq + i sinq, show that x(t) can be written as

x(t) = e^-bt ( B cos wt + C sin wt )

= D e^-bt cos(wt - j )

Where the intermediate step is shown as a hint on how to proceed. Show explicitly how D and j are related to A₁ and A₂. Note that in order for B, C, D and j to be real, both A₁ and A₂ must generally be complex.

Hardcopy entries should be submitted in class by noon on Monday of next week. Answers should be concise but complete.