
Dynamical systems
Suppose that $\displaystyle y' = ay  H$ $\displaystyle y(0) = y0$, models a fish population where a and H are positive constants and 0 <y0 < H/a. Find the time t* when the population dies out. [Hint: set y = 0 in solution formula (9) and solve for t.]
The solution formula, which I have confirmed both algebraically and through the answer book, is $\displaystyle y = y0e^[at]$
How can I solve for t in this problem when y = 0? $\displaystyle e^[at]$ never reaches zero. The only way I know to do this is to make y0 = 0. That does not solve for t, however.
The hint aside, I thought I would look for ways to make y' (the rate) = 0.
Therefore, y=H/a.
Where am I going wrong?
Thanks!

Hi there, I get a solution to the D.E. as
$\displaystyle \displaystyle y(t) = y_0 e^{at}\frac{H}{a} e^{at}+\frac{H}{a}$