The balancing transformation.

The problem with equation (9) is that the coefficients of

and

are not of the same order of magnitude in t, as

. A better choice of the second variable y when passing from the second-order equation (1) to an associated linear system (2) will remedy this. In general, equation (6) will be easier to analyze if the following two conditions are satisfied:

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 C onditions 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

and 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 3), 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 (11), with and .