Here is my derivation of the equation of motion for the damped driven harmonic oscillator.

Here is my playlist with the rest of the oscillator videos.
   • Harmonic Oscillators  

00:00 Intro
00:33 Review of Simple Harmonic Oscillator
02:00 Review of Damped Harmonic Oscillator
07:16 Under Damped Solution
09:34 Critically Damped and Over Damped
10:30 Damped Driven Oscillator
11:56 Characteristic and Particular Solution
13:49 Particular Solution
25:08 Full Solution
25:34 Review of Parameters
28:43 Review of Phase Space
29:54 Finding Velocity
31:49 Phase Space for Damped Driven Oscillator
32:34 Numerical Calculation