Case (i) If     there is one stationary solution .

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

            .  

The solution with the initial condition    is

            .
        
If    then   .

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

            .

Proof (i).

There is one real roots of the characteristic equation  ,  it is  .  

Thus there is one stationary solution  .

We will verify that    satisfies the D. E.  .   

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

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

Solution (i).

Now find the solution with the initial condition  .

(c) John H. Mathews 2004