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Rearranging the terms of the differential equation,  $2y\frac{dy}{dx}=x$. Now integrating on both sides,

$\Rightarrow$  $\int 2ydy=\int xdx$    $\Rightarrow$   $2\times (\frac{y^{2}}{2})= \frac{x^{2}}{2} +k$  i.e. $2y^{2}-x^{2}=C$

Option C is the answer.
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