 Solution

ID: 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

Let

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

, so that

y = x F (p)

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}) .

 New to topics? Read the docs here!