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160 4. The method of multiple scales

E4.1 A linear, damped oscillator
Given

with

we introduce

where the are constants, and write

where each is to be periodic in T and, we hope, uniformly valid as and as
(Note that the choice of T, in (4.5), follows exactly the pattern of a strained
coordinate; cf. Q3.24.) From (4.5) we obtain the operator identity

and so equation (4.3) becomes

where, as usual, ‘= 0’ means zero to all orders in When we insert (4.6) for X, we
obtain the sequence of equations