How are second order nonlinear ordinary differential equations solved?

I conceived the following second order nonlinear ordinary differential equation:

$$\frac{d^2y(x)}{dx^2}=\frac{k}{(y(x))^2}$$

I can tell it's nonlinear because of the $\frac{k}{(y(x))^2}$ term and second order because of the second order derivative.

Also, I did some research and concluded that it is of the type "missing x". In this category we use the relation, according to SOS math : $$\frac{d^2y(x)}{dx^2}=v\frac{dv}{dy} (1)$$ by making v equal to $$\frac{dy(x)}{dx} (2)$$ and then, using the chain rule, simplify to obtain equation (1). Despite all that, I can't seem to find a logical solution to my ODE. I would appreciate any help or clue! Thanks!

Typically, if your equation has a second derivative and a zeroth derivative but no first derivative, you can reduce the order by multiplying both sides by the first derivative and integrating. This works mathematically because the $y''$ gives you the "du" you need to integrate the left side, while the $y'$ gives you the "du" you need to integrate the right side. In many physical situations, this amounts to identifying a conservation law.

In this case you wind up with:

$$\frac{1}{2} y'^2=-\frac{k}{y}+C.$$

This equation is separable, provided you can unambiguously choose a sign for the square root.