A backwards funnel.

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A backwards funnel.

As is clear from Figure 9, all solutions of

with

enter and remain in the backwards funnel between

and

as

.

That is, for all t less than some negative constant . This inequality can be integrated from t to ; an argument identical to the one above then shows that

where

is the time when

enters the funnel. It can be shown [4, .6] that the solutions with

(more generally all solutions lying above the curve

in the left half-plane) become unbounded at a finite time

; such solutions

are defined only for

.

Exercise. Use similar techniques to identify the behavior of solutions to , for any k>0.

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