During the design of a car, part of the suspension system is tested by subjecting it to violent displacements. One such test is modelled by the differential equation

where x is displacement, and initially x = 0 and x = 0. The parameter k ( > 0) is known as the damping coefficient and can be varied during the tests. For optimum road holding a ‘hard’ suspension is desirable and it is believed that to achieve this the damping should be critical.
i) Find the value of k for critical damping.
ii) Determine x as a function of time t in this case. For a more comfortable ride a ‘soft’ suspension is proposed in which k = 0.6.
iii) Determine x as a function of time t for the ‘soft’ suspension.
iv) Find the maximum displacement of the ‘soft’ suspension.