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174 4. The method of multiple scales

This solution is oscillatory for Z > 0 and exponentially decaying for Z < 0; in par-
ticular

and

The equation for now becomes

and a particular integral of this equation is necessarily proportional to ZAi(Z), which
immediately leads to a non-uniformity in Thus we
mustselect

it is left as an exercise to show that, if we had written
then another non-uniformity would be present unless k = constant, and we have
already set k = 1. We alluded to this difficulty at the end of E4.5. Finally, the leading-
order solution will be completely determined once we have found h(X) (introduced
in (4.49)).
From (4.53) and (4.51), we have the equation

we consider the case X > 0, then it is convenient to write

This is

and so

which gives (with the appropriate choice of sign)