From this previous exercise (Section 8.7, Exercise #13) we have the population growth law

Find a formula that generalizes this equation in the case that the value of is a function of time rather than a constant. Express this result in terms of the time at which .

From our work in Exercise #13 (linked above) we know where is the unique solution of

Let

Then we apply Theorem 8.3 (page 310 of Apostol) with,

Therefore,

Thus,