Example 11.  Plot the solutions to the D. E.    in Example 9
that have the following initial conditions  .

Solution 11.

We could try the above technique where we solve    for  .
But it will involve complex numbers.  Hang in there !

Notice that there are multiple solutions for some of these values.
Special care and attention must be given to locate the ones to use, they are:

Now replace these constants.

Make them look nicer.  

It might be better to just solve the D.E. with the required initial conditions.

Then form the construction with the following two steps.

This is the same list that was obtained using those pesky complex numbers.

Graph these solutions.

Observe.  The family of functions
tend to the constant function    as  .  

(c) John H. Mathews 2004