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#### Analyzing after balancing.

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!

Contents Next: Analyzing r'/r. Up: The balancing transformation. Previous: The balancing transformation.