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in the which separate purely oscillatory from exponentially growing so-
lutions; see Further, for any given n, the asymptotic expansion will take the
form

and so we should then use the slow scale We note that the fast scale can be
taken simply as t. With these points in mind, we consider an example in some detail.

E4.4 Mathieu’s equation (n = 1) for
We use the method of multiple scales for the equation

with thus we have the case n = 1 and so we introduce the scales

Equation (4.39), with then becomes

where has been expanded in the usual way, and then we seek a solution

thus we generate the sequence of equations

and so on.
Equation (4.41a) has the general solution

and then (4.41b) can be written