7. (a) Find the general solution of the differential equation $$2 \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 5 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 2 x = 2 t + 9$$ (b) Find the particular solution of this differential equation for which \(x = 3\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = - 1\) when \(t = 0\). The particular solution in part (b) is used to model the motion of a particle \(P\) on the \(x\)-axis. At time \(t\) seconds \(( t \geq 0 ) , P\) is \(x\) metres from the origin \(O\).
(c) Show that the minimum distance between \(O\) and \(P\) is \(\frac { 1 } { 2 } ( 5 + \ln 2 ) \mathrm { m }\) and justify that the distance is a minimum.
(4)(Total 14 marks)