For any differential function y = f(x), the value of $\frac{d^2 y}{d x^2} + \left(\frac{dy}{d x}\right)^3 \frac{d^2 x}{d y^2}$ is :

always zero

$\left(\frac{d y}{d x}\right)=\left(\frac{d x}{d y}\right)^{-1}$ for a differential equation

or $\frac{d^2 y}{d x^2}=-1\left(\frac{dx}{dy}\right)^{-2} \frac{d}{d y}\left(\frac{d x}{d y}\right) \frac{d y}{d x}=-\left(\frac{dx}{dy}\right)^{-2} \frac{d^2 x}{d y^2} \frac{d y}{d x}=-\frac{d^2 x}{d y^2}\left(\frac{d y}{d x}\right)^3$

or $\frac{d^2 y}{d x^2}+\left(\frac{dy}{dx}\right)^3 \frac{d^2 x}{d y^2}=0$

Hence (1) is the correct answer.