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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**.

**Figure 5:** Fences 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 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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