I have the following question from a past paper which I seemed to be rusty on and just wanted to go through it:

A virus spreads slowly through a population of animals. Initially no animals are infected, but after 10 days 25% of the population in infected. Let

denote the proportion of the population that is infected. Then

(i) Solve the differential equation to get an expression for P(t).

(ii) Find the value of the constant k.

(iii) How long will it take for 75% of the population to become infected?

My attempt:

(i)

This seems wrong as at

, no animals should be infected, but the equation I derived gives a value of 2. What have I done wrong?

Also, why don't I have a coefficient in front of the

term? I have also never seen a constant next to the normal

term; is that normal?

I will wait till we have figured out part (i) before I attempt the next two parts.

Thanks.