Case (iii)  If   there are no stationary solutions.

When  ,  the differential equation has the form    and the solution is  

            .  

The solution with the initial condition    is

            .

The function  x(t)  has a vertical asymptote at   so the population   x(t)  becomes extinct at some time    (where .), i.e.  

            .

Proof (iii).

We will verify that    satisfies the D. E.  .   

Aside.  There are three equivalent forms for the solution for case (ii).

We can check out the limit as  .

To find the solution with the initial condition  ,  proceed as follows.

Example (iii).  Find the solution to the D. E.     using the general solution and compare it with the one found with Mathematica.

Solution (iii).

Aside.   Tan[z] = - Tan[-z].

Now find the solution with the initial condition  .

(c) John H. Mathews 2004