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

Question

Dr. Abubacker · Accepted Answer

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

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

Explanation

Reviewed by Dr. Abubacker

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