2. (a) Find the general solution of the differential equation $$t \frac { \mathrm {~d} v } { \mathrm {~d} t } - v = t , \quad t > 0$$ and hence show that the solution can be written in the form \(v = t ( \ln t + c )\), where \(c\) is an arbitrary constant.
(b) This differential equation is used to model the motion of a particle which has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\). When \(t = 2\) the speed of the particle is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find, to 3 significant figures, the speed of the particle when \(t = 4\).
[0pt] [P4 January 2002 Qn 6]