- The motion of a particle \(P\) along the \(x\)-axis is modelled by the differential equation
$$2 \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 5 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 2 x = 4 t + 12$$
where \(P\) is \(x\) metres from the origin \(O\) at time \(t\) seconds, \(t \geqslant 0\)
- Determine the general solution of the differential equation.
- Hence determine the particular solution for which \(x = 3\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = - 2\) when \(t = 0\)
- Show that, according to the model, the minimum distance between \(O\) and \(P\) is \(( 2 + \ln 2 )\) metres.
- Justify that this distance is a minimum.
For large values of \(t\) the particle is expected to move with constant speed.
- Comment on the suitability of the model in light of this information.