Find dy dx at x = 1, y = 4 if ^ 2 y + xy = K. - Vidyadip

Question

$\text{Find} \frac{\text{dy}}{\text{dx}} \text{at } x = 1, \text{y} = \frac{\pi}{4} \text{if } { \sin}^{2}\text{y} + \cos x\text{y = K}.$

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Answer

From the given equation
$2\sin \text{y } \cos \text{ y}. \frac{\text{dy}}{\text{dx}} - \sin \text{xy}.\bigg[\text{x}. \frac{\text{dy}}{\text{dx}} + \text{y . 1}\bigg] = 0$
$\Rightarrow \frac{\text{dy}}{\text{dx}} = \frac{\text{y} \sin \text{xy}}{\sin 2\text{y - x}\sin \text{(xy)}}$
$\therefore \frac{\text{dy}}{\text{dx}}\bigg|_{\text{x = 1, y =} \frac{\pi}{4}} = \frac{\pi}{4(\sqrt{2 - 1)}}$

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