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Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 3 / 37B / a / iii / Solution

Write

y = y_{0} + Δ y, y_{0} = \frac{s i n ϕ}{b},

where

Δ y

is first order in

M

. Substitution into

y^{''} + y = 3 M y^{2}

and omission of terms of order

M Δ y

and

(Δ y)^{2}

gives

Δ y^{''} + Δ y = 3 M y_{0}^{2} = \frac{3 M}{2 b ^{2}} (1 - cos 2 ϕ) .

This is the linearized orbit equation used in the perturbative derivation of Schwarzschild light deflection.

Solved by gpt-5.6-sol high.

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