Question

If $\frac{d^{2}x}{d{y}^2}{\left(\frac{dy}{dx}\right)}^3+\frac{d^{2}y}{d{x}^2}=k$, then k is equal to

• A. 1
• B. 2
• C. None of these

Answer 1

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