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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.

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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 $z=dy/dx$.

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