$2 x y d y = (x^{2} + y^{2}) d x .$
Solve the above equation.

x = k (x^{2} - y^{2})

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x^{3} = k (x^{2} + y^{2})

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x^{2} = k (x^{2} + y^{2})

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x^{2} = k (x^{3} - y^{3})

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Solution

The correct option is A $x = k (x^{2} - y^{2})$
$2 x y d y = (x^{2} + y^{2}) d x$
$\Rightarrow 2 \frac{d y}{d x} = \frac{x}{y} + \frac{y}{x}$
Substitute $y = v x \Rightarrow \frac{d y}{d x} = v + x \frac{d v}{d x}$
$\Rightarrow 2 v + 2 x \frac{d v}{d x} = v + \frac{1}{v}$
$\Rightarrow 2 x \frac{d v}{d x} = \frac{1}{v} - v = \frac{1 - v^{2}}{v}$
$\Rightarrow \frac{2 v}{1 - v^{2}} d v = \frac{1}{x} d x$
$\Rightarrow - log (1 - v^{2}) = log x - log k$
$\Rightarrow x (1 - v^{2}) = k \Rightarrow x^{2} - y^{2} = k x$

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