For the second system equation we differentiate (4), use the differential equation (1) to replace , and use equation (5) to replace :
For the Airy equation, with p=0 and q=t, this givesand the system of equations (2) becomes
We now choose and so as to satisfy our two conditions. The simplest choice for is for an appropriate n (where the negative sign makes the determinant of the resulting matrix positive). Condition I then implies
To satisfy Condition II, we need to ensure thatdecay at the same rate as . The first quantity reduces to if this is to decay at the same rate as , its largest power n+1 must be equal to -n, which implies that . This gives as announced in equation (10).
Exercise.Show that the variable is a balancing variable for the Bessel equationExercise.Find a balancing variable for the equation for any k>0, and deduce the asymptotic behavior of solutions as .