 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-1/2/a/solution

Remove the common nonzero factor

e^{- iω t}

and expand in the small parameter

ϵ

. To first order, the modified Laplacian contributes only its

y

x

derivative term has the extra factor

ϵ^{2}

. The regular perturbation expansion therefore gives

P = - iω b_{0} + ω ψ_{0, y}, Q = - iω b_{1} + ω ψ_{1, y} + ψ_{0, x} - b_{0, yy},

R = - iω ψ_{0, yy} - ω b_{0, y}, S = - iω ψ_{1, yy} - ω b_{1, y} - b_{0, x} .

Thus the two residuals are

P + ϵ Q + O (ϵ^{2})

and

R + ϵ S + O (ϵ^{2})

, respectively. Subscripts after a comma denote partial derivatives. We retain

ω \neq = 0

, as appropriate to the oscillatory wave and the elimination below.

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