2026 RVHS P1 Q6

A colony of bacteria is cultivated in a tank under experimental conditions. The amount of bacteria is \(y\) (in grams), at time \(t\) (in days) after the start of the experiment. The total growth rate of the bacteria is determined by the following two factors.

• The bacteria reproduce naturally and grow a rate directly proportional to the amount of bacteria present.
• An automated feeder continuously supplies a liquid nutrient solution to the tank. This nutrient solution stimulates an additional growth rate which is directly proportional to the time taken from the start of the experiment.

Initially, there is \(3\) grams of bacteria, and the total growth rate of the bacteria is \(6\) grams per day. After \(1\) day, the additional growth rate, due to the liquid nutrient, is \(3\) grams per day.

1. Show that \(\frac{\mathrm{d}y}{\mathrm{d}t}=2y+3t\).[2]
2. By substituting \(u=2y+3t\), show that the differential equation can be written as \(\frac{\mathrm{d}u}{\mathrm{d}t}=2u+3\).[2]
3. Find \(u\) in terms of \(t\) and hence \(y\) in terms of \(t\).[5]