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

Verify that y = x / (x + C) is a general solution for the given differential equation and show that all solutions contain (0,0) ...?

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.