Could someone explain how to reach an equation for this question, I've been struggling with it for a while.

Thanks

Rate of population increase is proportional to current population size can be expressed as

where k is some real constant.

Hence,

where k' = 1/k.

Then, antidifferentiating with respect to N, we have

(from context we can assume N > 0).

At this stage, before solving simultaneously for k' and c, it is convenient to write N in terms of t. This gives

We then use the facts that:

and

to solve simultaneously for k' and c in terms of initial population. This gives:

and

And substituting back into our equation for N, we have (with a bit of tidying up):

An alternative (much slicker) method is to begin by recognising that any solution to an equation of the form

will be a function of the form

Then substitute in known values (t = 0, f(0) = A, and t = 5, f(5) = 2A) to solve for b in terms of A. This gives

This agrees with the above approach, because b is the reciprocal of k'.