 Solution

ID: past-exam-of-the-perturbation-methods-and-applied-complex-variables-course-of-the-university-of-oxford/2008/a/4/i/solution

Treat

t, T

T = ϵ^{2} t

. The chain rule gives

\frac{d}{d t} = \partial_{t} + ϵ^{2} \partial_{T}, \frac{d ^{2}}{d t ^{2}} = \partial_{t}^{2} + 2 ϵ^{2} \partial_{t} \partial_{T} + ϵ^{4} \partial_{T}^{2} .

y_{tt} + y + ϵ cos t y + ϵ^{2} (2 y_{tT} + a_{2} y) + ϵ^{4} y_{TT} + O (ϵ^{4}) y = 0.

The unspecified

O (ϵ^{4})

term includes the higher correction in

a

. Only the displayed terms through order

ϵ^{2}

are needed to determine the leading slow-amplitude system.

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