Find dy dx if (x^ 2 + y^2)^ 2 = xy. - Vidyadip

Question

$\text{Find} \frac{\text{dy}}{\text{dx}} \text{if (x}^{2} + \text{y}^2)^{2} = \text{xy.}$

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Answer

$\text{(x}^{2} + \text{y}^{2})^{2} = \text{xy} \Rightarrow 2 \text{(x}^{2} + \text{y}^{2}) \bigg(\text{2x + 2y} \frac{\text{dy}}{\text{dx}}\bigg) = \text{x} \frac{\text{dy}}{\text{dx}} + \text{y}$

$\Rightarrow \text{4y} \frac{\text{dy}}{\text{dx}}\text{(x}^{2} + \text{y}^{2}) -\text{(x}^{2} + \text{y}^{2})- \text{x} \frac{\text{dy}}{\text{dx}} = \text{y - 4x} \text{(x}^{2} + \text{y}^{2})$

$\therefore \frac{\text{dy}}{\text{dx}} = \frac{\text{y - 4x}\text{(x}^{2} + \text{y}^{2})}{\text{4y}\text{(x}^{2} + \text{y}^{2}) - \text{x}}$

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