The motion of a damped spring-mass system (Figure) is described by the following ordinary differential equation:

where x = displacement from equilibrium position (m), t = time (s), m = 20-kg mass, and c = the damping coefficient (N ∙ s/m). The damping coefficient c takes on three values of 5 (under-damped), 40 (critically damped), and 200 (overdamped). The spring constant k = 20 N/m. The initial velocity is zero, and the initial displacement x = 1 m. Solve this equation using a numerical method over the time period 0 ≤ t ≤ 15 s. Plot the displacement versus time for each of the three values of the damping coefficient on the same curve.

d?x dx dt +c-+kx =D0