Example 13.  Discuss the graphs in the plot in Example 12.
What are the vertical lines ?
What are the curves that lie below x=1.
What are the curves that lie above x=3 ?  What use are they ?

Solution 13.

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.

Now form a composite graph of all the functions.

Observe.  The solutions with initial conditions  x[0] > 3  tend to   as .
The solutions with initial conditions  1 < x[0] < 3  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 functions    and    are the dividing lines for these three cases.

(c) John H. Mathews 2004