Question

Solve: p² + 2py cot x = y².

Answer 1

The given equation is 

p² + 2py cot x = y² ...(i)

(p + y cot x)² = y²(1 + cot²x)

(p + y cot x)² = y²cosec² x

(p + y cot x) = ± y cosec x

p = y(cosec x – cot x) ...(ii)

and p = – y(cosec x – cot x) ...(iii)

From Eq. (ii), we get, 

dy/dx = y(cosec x – cot x)

dy/y = (cosec x – cot x)dx

dy/y = ((1 – cos x)/sin x)dx 

dy/y = tan(x/2) dx

Integrating, we get