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Here we show the proofs alluded to in the section on analyzing
, after the balancing transformation.
Theorem 4.7.5 of [4, p. 188] asserts that if is a
narrowing antifunnel for the differential equation , and if
there is a function such that
,
in U,
and such that , then there is a unique
solution remaining in U as .
Applied to equation (12), we find
for t > 0, so with ,
the uniqueness criterion is satisfied. This shows that there is a unique
solution remaining in each antifunnel.
To see that any solution of equation (12)
does stay in one of these antifunnels, let and
be defined respectively by the properties
Conceivably or , but certainly , and
we will be done if we can show that .
In Figure 14,
we redraw
Figure 5 using and .
If , choose C with ; the unique solution
lying in the antifunnel defined by and
must be either above or beneath it, because
solutions cannot intersect. If it is above, is also above,
contradicting the definition of ; if it is beneath, then so is
, contradiciting the definition of . Thus, every solution
is of the form for some C, and there is a unique such
solution for each C.
Figure 14: Fences and
associated with a solution .
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