9 The quantity of grass on an island at time \(t\) years is \(x\), in appropriate units. At time \(t = 0\) some rabbits are introduced to the island. The population of rabbits on the island at time \(t\) years is \(y\), in units of 100s of rabbits.
An ecologist who is studying the island suggests that the following pair of simultaneous first order differential equations can be used to model the population of rabbits and quantity of grass for \(t \geqslant 0\).
$$\begin{aligned}
& \frac { \mathrm { d } x } { \mathrm {~d} t } = 3 x - 2 y ,
& \frac { \mathrm {~d} y } { \mathrm {~d} t } = y + 5 x
\end{aligned}$$
- (a) Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = a \frac { \mathrm {~d} x } { \mathrm {~d} t } + b x\) where \(a\) and \(b\) are constants which should be found.
(b) Find the general solution for \(x\) in real form. - Find the corresponding general solution for \(y\).
At time \(t = 0\) the quantity of grass on the island was 4 units. The number of rabbits introduced at this time was 500 .
- Find the particular solutions for \(x\) and \(y\).
- The ecologist finds that the model predicts that there will be no grass at time \(T\), when there are still rabbits on the island.
Find the value of \(T\).
- State one way in which the model is not appropriate for modelling the quantity of grass and the population of rabbits for \(0 \leqslant t \leqslant T\).
\section*{OCR}
\section*{Oxford Cambridge and RSA}