2022 MI J2 MYE P1 Q9

A team of scientists is studying the change in population of a particular species of fish in a lake. It is observed that the rate of change of the population is proportional to the product of the population and the difference between a constant \(a\) and the population. If the population of this fish at time \(t\) years is \(x\) thousand, it is found that the rate of change of the population is \(400\) fish per year when the population is \(1000\) and \(600\) fish per year when the population is \(3000\).

1. Show that \(\frac{\mathrm{d}x}{\mathrm{d}t}=\frac{1}{10}x\left( 5-x \right)\).[3]
2. Find the general solution of the differential equation in part (i), giving your answer in the form \(x=\mathrm{f}\left( t \right)\).[5]