Homework Help: Harvesting Model

Here is the question:

Investigate the harvesting model in problem 5 both qualitatively and analytically in the case a=5, b=1, and h=25/4. Determine whether the population becomes extinct in finite time. If so, find that time.

The information from problem 5 is:

dP/dt= P(a-bP)-h , P(0)=P₀


Attempt:
I subbed in the values for a,b, and h into the problem. Used variable separable and integrated both sides. My solution to the integration came out to be:

-2/(2P-5) = -t + C

I have absolutely no clue where to go from there. My roommate who took the class last semester couldn't figure out what to do either. Any help would be much appreciated.

 

by solving for P, I get: P= 1/t + 5/2. With what respect am I taking the derivative because if you do P you will get 1. Otherwise dP/dt= -1/t². I'm still stuck either way because our professor doesn't explain topics very well. He just writes things out and doesn't actually say what he did so following it isn't easy.

 

I guess I forgot to write the C in. but I've come up with something else but I don't know if it's correct.

Taking P= 5/2 + 1/(C+t) I used P(0)=P₀ = 1/C +5/2. then solved for C and got.
C=1/(P₀-(5/2))
Then: used P(t₁)=0 and plugged in my C value into the original P= 5/2 + 1/(C+t) getting:
5/2 +1/(1/P₀-(5/2))+t₁)=0. I then solved for t getting
t₁= (2/5)(-5/(2P₀+5)-2/5.

My next thought I had was to do: lim (t->infinity)P(t)= 5/2 + 1/(1/P₀-(5/2))+t₁). When I take the limit my answer for P(t)= 5/2. Im not sure what that answer actually gives me.