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 Incoming links: Slope parameter for a homogeneous first-order equation

Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2016 / Paper 2 / 14 / c / i / Solution 2026-10-18

Let

F (p) = 2 p / (1 - p^{2})

, so that

y = x F (p)

. Differentiation gives

p = \frac{2 p}{1 - p ^{2}} + x \frac{2 ( 1 + p ^{2} )}{( 1 - p ^{2} ) ^{2}} \frac{d p}{d x} .

Move the first term to the left and use

p - \frac{2 p}{1 - p ^{2}} = - \frac{p ( 1 + p ^{2} )}{1 - p ^{2}} .

Since

1 + p^{2} > 0

for real

p

and

p \neq = \pm 1

, cancellation gives

2 x \frac{d p}{d x} = - p (1 - p^{2}) .

This is the slope parameter for a homogeneous first-order equation reduction.

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