5. An elastic string spring of modulus \(2 m g\) and natural length \(l\) is fixed at one end. To the other end is attached a mass \(m\) which is allowed to hang in equilibrium. The mass is then pulled vertically downwards through a distance \(l\) and released from rest. The air resistance is modelled as having magnitude \(2 m \omega v\), where \(v\) is the speed of the particle and \(\omega = \sqrt { \frac { g } { l } }\). The particle is at distance \(x\) from its equilibrium position at time \(t\).

1. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \omega \frac { \mathrm {~d} x } { \mathrm {~d} t } + 2 \omega ^ { 2 } x = 0\).
2. Find the general solution of this differential equation.
3. Hence find the period of the damped harmonic motion.