1. A scientist is studying the effect of introducing a population of type \(A\) bacteria into a population of type \(B\) bacteria.

At time \(t\) days, the number of type \(A\) bacteria, \(x\), and the number of type \(B\) bacteria, \(y\), are modelled by the differential equations $$\begin{aligned} & \frac { \mathrm { d } x } { \mathrm {~d} t } = x + y
& \frac { \mathrm {~d} y } { \mathrm {~d} t } = 3 y - 2 x \end{aligned}$$

1. Show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } - 4 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 5 x = 0$$
2. Determine a general solution for the number of type \(A\) bacteria at time \(t\) days.
3. Determine a general solution for the number of type \(B\) bacteria at time \(t\) days. The model predicts that, at time \(T\) hours, the number of bacteria in the two populations will be equal. Given that \(x = 100\) and \(y = 275\) when \(t = 0\)
4. determine the value of \(T\), giving your answer to 2 decimal places.
5. Suggest a limitation of the model.