Example 6.  Discuss the graphs in the plot in Example 5.
What are the vertical lines ?
What are the curves that lie below x=1.
What are the curves that lie above x=1 ?  What use are they ?

Solution 6.
In order to clear things up, it is necessary to specify the individual domain, for each of the solutions, then plot all the curves on the same graph.

Recall from example 4 that the constant solution is  .  

The vertical lines are asymptotes of the solutions .  

The curves that lie below are the solutions with the initial conditions we wanted.

The curves that lie above are extraneous portions of the solution functions.   
They are not useful for our purposes.  In order to get rid of them it will take some effort.

Now form a composite graph of all the functions.

Observe.  The solutions with initial conditions  x[0] > 0  tend to   as .
The solutions with initial conditions  0 < x [0] < 1  go to zero at some finite value of t and  x[t]  becomes extinct.
The constant function    is the dividing line for these two cases.

(c) John H. Mathews 2004