# Non-linear differential equation I

What is the solution to the non-linear differential equation $$\frac{d^2 y}{dx^{2}} = \left( \frac{2 y -1}{y^2 + 1} \right) \, \left( \frac{dy}{dx} \right)^2\ \text{ ?}$$ I would suspect it has a connection to the inverse tangent function, but the trick of getting there seems to be elusive.

Let $z=dy/dx$ then we can rewrite the equation as,
$$\frac{dz}{dx} = \frac{2y-1}{y^2+1} z^2,$$
and using the chain rule we can write $dz/dx = dy/dx \times dz/dy = z\times dz/dy$ , $$z \frac{dz}{dy} = \frac{2y-1}{y^2+1} z^2,$$
$$\int_{z_0}^z \frac{dz'}{z'} = \int_{y_0}^y \frac{2y'-1}{y'^2+1} dy'.$$
Once we have $z(y)$ from the above we can obtain $y(x)$ by integration of the equation $z=dy/dx$.