N2009 P2 Q4

Two scientists are investigating the change of a certain population of size \(n\) thousand at time \(t\) years.

1. One scientist suggests that \(n\) and \(t\) are related by the differential equation \(\frac{{{\mathrm{d}}^{2}}n}{\mathrm{d}{{t}^{2}}}=10-6t\). Find the general solution of this differential equation. Sketch three members of the family of solution curves, given that \(n=100\) when \(t=0\).[5]
2. The other scientist suggests that \(n\) and \(t\) are related by the differential equation \(\frac{\mathrm{d}n}{\mathrm{d}t}=3-0.02n\). Find \(n\) in terms of \(t\), given again that \(n=100\) when \(t=0\). Explain in simple terms what will eventually happen to the population using this model.[7]