Find the general solution of the differential equation \(t \frac { \mathrm {~d} v } { \mathrm {~d} t } - v = t , t > 0\) and hence show that the solution can be written in the form \(v = t ( \ln t + c )\), where \(c\) is an arbitrary cnst.
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 sf , the speed of the particle when \(t = 4\).