 Solution

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

At order one,

(\partial_{t}^{2} + 1) y_{0} = 0

, so

y_{0} = A (T) cos t + B (T) sin t .

At order

ϵ

(\partial_{t}^{2} + 1) y_{1} = - cos t y_{0} = - \frac{A}{2} - \frac{A}{2} cos 2 t - \frac{B}{2} sin 2 t .

The operator sends a constant to itself and sends

cos 2 t, sin 2 t

to minus three times themselves. Hence

y_{1} = - \frac{A}{2} + \frac{A}{6} cos 2 t + \frac{B}{6} sin 2 t .

An additional homogeneous

cos t, sin t

contribution can be absorbed into the definitions of

A, B

. There is no resonance forcing at this order; the first amplitude evolution therefore occurs on the chosen slower scale

T = ϵ^{2} t

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