Derive the exact ODE solved by specific function

Question

I have been given the task to find the exact ODE which is solved by \begin{equation} y(x)=\pm\sqrt{K|x|-1} \end{equation} I ended up with $y'=\pm \frac{K}{2y}$, but I am not sure that this is correct. I was then given the task to derive this ODE and check whether my result was correct by inserting the original equation into the new ODE that I derived. I do not know what to do from here. Any help would be appreciated.

Answer 1

\begin{align} \qquad & y^2 = K|x| - 1 \\ \implies &y^2 + 1 = K|x|\quad \left(\implies K > 0, |x| \geq \frac{1}{K}\right) \\ \implies &2yy' = K\operatorname{sgn}(x) \\ \implies &2yy' = K\left(\frac{x}{|x|}\right) = (y^2+1) \frac{x}{|x|^2}\\ \implies &2xyy' = (y^2 + 1),\; \text{ given that } x \neq 0. \end{align} The above equation is the required ODE, with the necessary domain conditions.