How do I eliminate the arbitrary constant in the equation y = ax^2 -1?

The answer at the back of the book is 2y = x(dy/dx)-2 however I don't understand how they arrived at this.

Here is my attempt:

y = ax^2 -1

dy/dx = 2ax

(dy/dx) (1/2x) = a

therefore y = (dy/dx) (1/2x)x^2 - 1

Where did I go wrong?