 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/b/solution

Insert the proposed leading relation into the two zeroth-order residuals. It gives

P = - iω (- i ψ_{0, y}) + ω ψ_{0, y} = 0,

and, since

b_{0, y} = - i ψ_{0, yy}

R = - iω ψ_{0, yy} - ω (- i ψ_{0, yy}) = 0.

Hence

b_{0} = - i ψ_{0, y} ⟹ P = R = 0

. The same relation is forced by

P = 0

when

ω \neq = 0

; its

y

derivative then automatically gives

R = 0

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