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Analyzing for the Airy equation.

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If the Airy equation is transformed into a system of the form (2) in the standard way, setting , we obtain the linear time-dependent system

 


The Prüfer transformations are then applied to the Airy equation, as represented by system (8), equation (6) becomes

 

Trajectories of this equation are shown in Figure gif. In the left half-plane, various funnels and antifunnels are visible, but they do not give any better information than the Ricccati equation (16). The solutions on the right of Figure gif seem to be merging (far offscreen) as , but they are too wiggly for us to be able to find good fences for a funnel or antifunnel argument.

  
Figure 2: Trajectories for equation for , before balancing.



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Contents Next: The balancing transformation. Up: The Prüfer Transformation Previous: The Prüfer Transformation