Mathematics - Differential Equation Question with Solution | TestHub

$\frac{dy}{dx}={\left(\frac{dx}{dy}\right)}^{-1}$ $\Rightarrow \frac{d^{2}y}{d{x}^2}=-1{\left(\frac{dx}{dy}\right)}^{-2}\left\{\frac{d}{dx}\left(\frac{dx}{dy}\right)\right\}$ $\Rightarrow \frac{d^{2}y}{d{x}^2}=(-1){\left(\frac{dx}{dy}\right)}^{-2}\left\{\frac{d}{dy}\left(\frac{dx}{dy}\right)\frac{dy}{dx}\right\}$ $=(-1){\left(\frac{dy}{dx}\right)}^2\left\{\frac{d^{2}x}{d{y}^2}\cdot \frac{dy}{dx}\right\}=-{\left(\frac{dy}{dx}\right)}^3\left\{\frac{d^{2}x}{d{y}^2}\right\}$ $\Rightarrow \frac{d^{2}x}{d{y}^2}{\left(\frac{dy}{dx}\right)}^3+\frac{d^{2}y}{d{x}^2}=0$