after balancing.
,
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 5.
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.
and 
forming the antifunnel 
for equation
.
These facts allow us to determine the spacing of the zeroes of 
.
Recall that 
,
so 
if and only if 
is an odd multiple of 
.
Figure 6 shows the
t-intervals in which the zeros of 
must lie;
that is, the intervals between the curves 
and 
corresponding to equally spaced values of 
.
Figure 6: The t-intervals trapping the zeroes of 
; here
.
Let 
denote the zero of 
correponding to 
. Since the solution 
stays inside the
antifunnel 
, it is clear that the interval defining 
shrinks
as 
; in fact, for large n we have
Hence (l'Hôpital) the distance 
between successive zeros
approaches
0 as 
. So our intuition that the oscillations are more and
more compressed, and that the distance between successive zeros goes to
zero as 
, was correct!
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