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As is clear from Figure , 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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**Up:** The Riccati Transformation
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