 Solution

ID: past-exam-of-the-perturbation-methods-and-applied-complex-variables-course-of-the-university-of-oxford/2010/a/3/i/solution

Matching powers of

ϵ

gives

u_{0}^{''} + u_{0} = 0

u_{1}^{''} + u_{1} = u_{0}^{2}

, and

u_{2}^{''} + u_{2} = 2 u_{0} u_{1}

, with

u_{0} (0) = 2

and homogeneous initial data for every higher correction. Thus

u_{0} = 2 cos t, u_{1} = 2 - \frac{4}{3} cos t - \frac{2}{3} cos 2 t .

There is no resonant fundamental forcing at first order. At second order,

2 u_{0} u_{1} = \frac{20}{3} cos t - \frac{8}{3} - \frac{8}{3} cos 2 t - \frac{4}{3} cos 3 t .

The first term resonates with the homogeneous oscillator. Solving with the initial data gives

u_{2} = \frac{10}{3} t sin t - \frac{8}{3} + \frac{29}{18} cos t + \frac{8}{9} cos 2 t + \frac{1}{6} cos 3 t .

u = u_{0} + ϵ u_{1} + ϵ^{2} u_{2} + O (ϵ^{3})

for fixed

t

. The first secular term is $(10/3) ϵ^{2} t sin t$ , at order $ϵ^{2}$ . It makes the regular expansion nonuniform on times of order

ϵ^{- 2}

 New to topics? Read the docs here!