Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2025 / Paper 2 / 13 / a / i

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- Solution i

Solution

Differentiate the second ordinary differential equation and use the first to eliminate

x^{'}

:

y^{''} = x^{'} - sin t = β y + sin t - sin t .

Thus the reduced second-order linear differential equation is

y^{''} - β y = 0 .

No solutions are lost: given a solution

y

x = y^{'} - cos t

; then both original ordinary differential equations hold.

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