Case (ii)  If   there are two stationary solutions    and  .

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

            .

The solution with the initial condition    is

    .

If      then   .  
    
If      then the population   x(t)  becomes extinct at some time  ,  i. e.   .  

Proof (ii).

The two real roots of the characteristic equation  ,  are  .  

Thus there are two stationary solutions    and  .

We will verify that    satisfies the D. E.  .   

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

This third form is the one we will promote.

It is difficult for Mathematica to find the limit of  x[t]  because of the arbitrary constants (which are assumed to be complex symbols in Mathematica).

So we will need to be clever in taking the limit.

This seems to verifies the statement

        If     then  .  

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

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

Solution (ii).

Now find the solution with the initial condition  .

(c) John H. Mathews 2004