4. A particle \(P\) of mass \(m\) is attached to the mid-point of a light elastic string, of natural length \(2 L\) and modulus of elasticity \(2 m k ^ { 2 } L\), where \(k\) is a positive constant. The ends of the string are attached to points \(A\) and \(B\) on a smooth horizontal surface, where \(A B = 3 L\). The particle is released from rest at the point \(C\), where \(A C = 2 L\) and \(A C B\) is a straight line. During the subsequent motion \(P\) experiences air resistance of magnitude \(2 m k v\), where \(v\) is the speed of \(P\). At time \(t , A P = 1.5 L + x\).

1. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 k \frac { \mathrm {~d} x } { \mathrm {~d} t } + 4 k ^ { 2 } x = 0\).
2. Find an expression, in terms of \(t , k\) and \(L\), for the distance \(A P\) at time \(t\).