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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

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

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
** Exercise.**Show that the variable
is a balancing variable for the Bessel equation

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