- Two species of insects, \(X\) and \(Y\), co-exist on an island. The populations of the species at time \(t\) years are \(x\) and \(y\) respectively, where \(x\) and \(y\) are measured in millions. The situation can be modelled by the differential equations
$$\begin{aligned}
& \frac { \mathrm { d } x } { \mathrm {~d} t } = 3 x + 10 y
& \frac { \mathrm {~d} y } { \mathrm {~d} t } = x + 6 y
\end{aligned}$$
- Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } - 9 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 8 x = 0\).
- Find the general solution for \(x\) in terms of \(t\).
- Find the corresponding general solution for \(y\).
- When \(t = 0 , \frac { \mathrm {~d} x } { \mathrm {~d} t } = 5\) and the population of species \(Y\) is 4 times the population of species \(X\). Find the particular solution for \(x\) in terms of \(t\).