Negative order differential equations Are algebraic solutions for a function y(x) fulfilling this differential equation possible? \begin{equation} \text{y}^{-2}= \frac{\text{d}^2\text{y}}{\text{dx}^2} \end{equation} I found this differential equation, in the context of motion due to a fixed, repelling force that follows an inverse square law. I figured out the following properties of y, when searching for numerical solutions, where i expect the algebraic solutions to show these features: With given initial condition $\text{y} > 0$: \begin{equation} \lim_{x\to \infty} \frac{\text{dy}}{\text{dx}} = \text{constant} \end{equation} I also know that the function is not going to "behave well" for $\lim_{y\to 0}$, due to its second derivative going to infinity in that case.

**Just a hint**

The differential equation can be written as

$$\frac {y'}{y^2}=y' y''$$

and after integration

$$\frac {-1}{y}=\frac {1}{2}(y')^2+C$$