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#### Choosing the balancing variable.

Given the equation

we will now see how to choose a second variable

so that Conditions I and II are satisfied. We first need to find the system of first-order equations satisfied by x and y; this is old-fashioned computation, but not difficult. The first equation simply comes from solving (4) for to give

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 gives

and 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 that

decay 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 equation

Exercise.Find a balancing variable for the equation for any k>0, and deduce the asymptotic behavior of solutions as .

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