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The Riccati substitution transforms the linear time-dependent system (2) into a first-order nonlinear ordinary differential equation in the single variable

**Figure 8:** Trajectories for the Riccati equation
associated with the Airy equation.

It is clear from Figure that the left half-plane **t < 0**,
corresponding to the region of the **xt**-plane in Figure where
trajectories seem to be non-oscillatory, is more amenable to analysis,
because solutions appear to funnel together or spray apart. In fact, this
is a mirror image (about the **u**-axis) of the phase plane analyzed by
John Hubbard in [3]. Since all of this deals with
question 2, we shall defer further discussion until we have dealt with
question 1: the behavior of the solutions for **t > 0**. The oscillatory
solution for **t>0** shown in Figure suggests that we try the Prüfer
transformation.

We redraw Figure
as Figure , adding isoclines of slopes **-1, 0, +1**.
Portions of these
three isoclines act as fences to form funnels and antifunnels.

**Figure 9:** Trajectories for the Riccati equation , with isoclines of slopes **-1, 0, +1**.

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