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 beand system (2) becomes We will explain in Appendix B how (10) and (11) are found, and how to find balancing variables for other equations. An interesting exercise for the student is to check that the system (11) is indeed equivalent to the second-order Airy equation .
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 :which yields a graph (Figure ) to compare with Figure ().
Figure 4: Trajectories for equation for , after balancing.