The solutions on the right of Figure
are merging (far offscreen) as ,
and they are ``straight'' enough to try using the
theory of fences and funnels.
In fact, as a result of Condition II, we see from equation (12) that as
, solutions are going to behave very much like solutions to
the equation . To be precise, from (12) we get
for all t > 0. Therefore, for any C, the curves
are respectively a lower fence and an upper fence for the differential
equation (12). That is, for every t>0 the slope
the slope of the direction field for at that point, and
the slope of the direction field. Together these fences
define an antifunnel
, which is narrowing as ; see Figure .
We will show in Appendix C that there is exactly one solution
which is trapped in the narrowing antifunnel for all
t>0, and that every solution of (12) for t > 0 is of this form for
some C.