Although is the usual choice of a second variable when going from
a second-order equation to a first-order system, it is perfectly
permissible to let y be any linear combination of x and
:
. If we can choose the coefficient functions
and
so that Conditions I and II are satisfied, we will
call the resulting y a balancing variable.
For the Airy equation, a balancing variable is found to be
When we look at the graphical results (Figure ),
we can see that the
trajectory in the phase plane for this system is more ``circular''
(i.e.,
x and y are ``balanced''), than in Figure 1.
Figure 3: Graphs of solution to balanced system (), with and
.
This is reflected in the resulting equation (6) for :
Figure 4: Trajectories for equation for , after balancing.