Substitution in Differential Equation

Calculus Level pending

\[\dfrac{d^2y}{dx^2}= 1+ \dfrac{2(1+y)}{1+y^2}\left(\dfrac{dy}{dx}\right)^2\] Let \(y=\tan z\). Then the differential equation above becomes \[\dfrac{d^2z}{dx^2}= \cos^2z+ k\left(\dfrac{dz}{dx}\right)^2\] for some constant \(k\). Find the value of \(k\).

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