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with for and with suitable initial conditions. The most natural
and convenient formulation of the multiple-scale problem follows E4.3: we introduce

and seek a solution

Equation (4.44) becomes

and, in this first exercise, we will not expand (but see Q4.15). Thus (4.45) leads to
the sequence of equations

for the O(1) and problems.
To solve (4.46a), we first select and then obtain the general solution

where and are arbitrary functions. (Other choices of lead to a formulation in
terms of but then the period will depend on which leads to additional non-
uniformities—see E4.6—unless is constant; we have made the simplest choice of
this constant.) The equation for now becomes

where we have written The solution for is periodic in T, i.e. in
if

and so

So, for example, if we are given the initial data then we have