The ODE mx'' + cx' + kx = 0 is used to model a damped mass-spring system, where m is the mass, c is the damping constant and k is the spring constant. Describe and explain the behaviour of the system for the cases: (a) c^2>4mk; (b) c^2=4mk; (c) c^2<4mk.

In the case c²>4mk, the characteristic equation has two distinct real roots; this represents overdamping. The system does not oscillate, and x approaches zero as time approaches infinity.In the case c²=4mk, the characteristic equation has a repeated real root; this represents critical damping. The system does not oscillate and returns to its equilibrium position in the shortest possible time; x approaches zero as time approaches infinity.In the case c²<4mk, the characteristic equation has two complex routes; this represents underdamping. The system oscillates with an exponentially decreasing amplitude; the amplitude of oscillations approaches zero as time approaches infinity.