# Dynamical systems

• January 23rd 2011, 06:49 PM
Truthbetold
Dynamical systems
Suppose that $y' = ay - H$ $y(0) = y0$, 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 $y = y0e^[at]$

How can I solve for t in this problem when y = 0? $e^[at]$ 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!
• January 23rd 2011, 07:17 PM
pickslides
Hi there, I get a solution to the D.E. as

$\displaystyle y(t) = y_0 e^{at}-\frac{H}{a} e^{at}+\frac{H}{a}$