$(x^{2} - y^{2}) d x + 2 x y d y = 0$ , the solution to this differential equation represents which curve:

line

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circle

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parabola

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ellipse

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Solution

The correct option is C circle
$\frac{d y}{d x} = \frac{y^{2} - x^{2}}{2 x y}$
Substitute $y = v x$
$∴ v + x \frac{d v}{d x} = \frac{v^{2} - 1}{2 v} \Rightarrow x \frac{d v}{d x} = - \frac{1 + v^{2}}{2 v} \Rightarrow \frac{2 v}{1 + v^{2}} d v + \frac{d x}{x} = 0$
Integrating $l o g (1 + v^{2}) + l o g x = l o g k$
$∴ x (1 + v^{2}) = k .$
Substitute $v = \frac{y}{x}$
$\Rightarrow x^{2} + y^{2} = k x .$

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(1, 1)

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