1. The spread of a rumour in a community can be modelled by the differential equation

dp / dt = f(p)(1 ? p),

(1) where p represents the proportion of a population that has heard the rumour at time t (days), and f(p) is some function of the population p.

(a) Suppose the rumour spreads at a constant rate so that f(p) = k, where k > 0. This means we have the differential equation dp / dt = k(1 ? p).

If 1000 people live in this community, and initially, 5 people have heard this rumour, find the particular solution. What proportion of the population will eventually hear this rumour?

(b) Let us now consider a different model where the spread of the rumour occurs when people who have heard the rumour meet those who have not heard the rumour. This means f(p) = rp, where r > 0.

i. Using this new form for f(p), find the general solution to equation (1). You may leave the solution in implicit form.

ii. If 5 people have initially heard this rumour, from the total population of 1000, show that the particular solution for this model is