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What Happens for Negative t?

Examining the graphs of the solutions of the Airy equation for t<0, shown in Figure 7, we observe very different behavior from what was shown for positive t in Figure 1. Note that in Figure 7 the solutions either increase or decrease without bound as t heads toward negative infinity.

Figure 7: Solutions of the Airy equation for t<0, with and various values for .

It is easy to see that there is no oscillatory behavior, but are there any solutions that are bounded? An interesting computer exercise for the student is to try varying the initial values for the velocity , while holding fixed, in order to find an experimental value for that gives a bounded solution. (We selected to make the experiment more interesting, since gives the trivial solution.) Our computer experiments indicate that if , the solution appears to remain bounded as t becomes large negative. To find out what is really going on, we turn to the Riccati equation associated with the Airy equation .

Contents Next: The Riccati Transformation Up:A New Look at Previous: Analyzing r'/r.