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The solutions on the right of Figure 4
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 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.
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 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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Next: Analyzing r'/r.
Up: The balancing transformation.
Previous: The balancing transformation.