1. A patient is treated by administering an antibiotic intravenously at a constant rate for some time.

Initially there is none of the antibiotic in the patient.
At time \(t\) minutes after treatment began

• the concentration of the antibiotic in the blood of the patient is \(x \mathrm { mg } / \mathrm { ml }\)
• the concentration of the antibiotic in the tissue of the patient is \(y \mathrm { mg } / \mathrm { ml }\)

The concentration of antibiotic in the patient is modelled by the equations $$\begin{aligned} & \frac { \mathrm { d } x } { \mathrm {~d} t } = 0.025 y - 0.045 x + 2
& \frac { \mathrm {~d} y } { \mathrm {~d} t } = 0.032 x - 0.025 y \end{aligned}$$

1. Show that $$40000 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + 2800 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 13 y = 2560$$
2. Determine, according to the model, a general solution for the concentration of the antibiotic in the patient's tissue at time \(t\) minutes after treatment began.
3. Hence determine a particular solution for the concentration of the antibiotic in the tissue at time \(t\) minutes after treatment began. To be effective for the patient the concentration of antibiotic in the tissue must eventually reach a level between \(185 \mathrm { mg } / \mathrm { ml }\) and \(200 \mathrm { mg } / \mathrm { ml }\).
4. Determine whether the rate of administration of the antibiotic is effective for the patient, giving a reason for your answer.