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Then, with conditions (5.63), we see that

which leaves the solution for as

It is left as an exercise to show that the solution of equation (5.61c), for is
periodic in T and bounded as if

with and

This example has proved to be a particularly straightforward application of the method
of multiple scales; the next is a rather less routine problem that contains a turning point
(see E4.6).

E5.9 Planetary rings
In a study of a model for differentially rotating discs (Papaloizou & Pringle, 1987), the
radial structure of the azimuthal velocity component for large azimuthal mode number
(essentially here) satisfies an equation of the form

This equation clearly possesses a turning point at (see §2.8); let us examine the
solution near this point first. For the neighbourhood of we set
where as and ignore the scaling of v (because the equation is
linear), so equation (5.64) becomes (with

Thus we select and with we obtain the leading-order
equation

which is an Airy equation (see equations (4.52), (4.54)) with a bounded solution