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'.