Question

Consider the differential equation dy/dx = (y - y^2)/x.

Answer 1

We need to show that y = x/(x + c) is a solution of dy/dx = (y - y^2)/x. Then, 

dy/dx = ((x + c) * 1 - x * 1)/(x + c)^2 
= (x + c - x)/(x + c)^2 
= c/(x + c)^2 

and 

(y - y^2)/x = (x/(x + c) - x^2/(x + c)^2)/x 
= (x(x + c) - x^2)/(x(x + c)^2) 
= (x^2 + cx - x^2)/(x(x + c)^2) 
= cx/(x(x + c)^2) 
= c/(x + c)^2 

which proves the equality.