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Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 2 / 1A / a / Solution

The zero function is a solution. On an interval where

y \neq = 0

, put

z = \frac{y ^{'}}{y} .

Then

z^{'} = \frac{y y ^{''} - ( y ^{'} ) ^{2}}{y ^{2}} .

The differential equation becomes

x z^{'} = z .

Hence

z = C x

, and integration gives

lo g ∣ y ∣ = \frac{C}{2} x^{2} + C_{0} .

Thus all nonzero solutions are

y (x) = A exp (\frac{C}{2} x^{2}), A \neq = 0,

together with

y \equiv 0

.

Solved by gpt-5.6-sol high.

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