A particle \(P\) of mass 4 kg moves along a horizontal straight line under the action of a force directed towards a fixed point \(O\) on the line. At time \(t\) seconds, \(P\) is \(x\) metres from \(O\) and the force towards \(O\) has magnitude \(9 x\) newtons. The particle \(P\) is also subject to air resistance, which has magnitude \(12 v\) newtons when \(P\) is moving with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Show that the equation of motion of \(P\) is
$$4 \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 12 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 9 x = 0$$
It is given that the solution of this differential equation is of the form
$$x = \mathrm { e } ^ { - \lambda t } ( A t + B )$$
When \(t = 0\) the particle is released from rest at the point \(R\), where \(O R = 4 \mathrm {~m}\).
Find,
the values of the constants \(\lambda , A\) and \(B\),
the greatest speed of \(P\) in the subsequent motion.