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The solution of the differential equation, $x (x^{2} + 3 y^{2}) d x + y (y^{2} + 3 x^{2}) d y = 0$ is

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$x^{4} + y^{4} + x^{2} y^{2} = c$

$x^{4} + y^{4} + 3 x^{2} y^{2} = c$

$x^{4} + y^{4} + 6 x^{2} y^{2} = c$

$x^{4} + y^{4} + 9 x^{2} y^{2} = c$

answer is C.

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Detailed Solution

$x (x^{2} + 3 y^{2}) d x + y (y^{2} + 3 x^{2}) d y = 0$

$\Rightarrow \frac{4 x^{3} d x + 4 y^{3} d y + 12 x y^{2} d x + 12 x^{2} y d y}{x^{4} + y^{4} + 6 x^{2} y^{2}} = 0 \Rightarrow \ln (x^{4} + y^{4} + 6 x^{2} y^{2}) = \ln c$

$\Rightarrow x^{4} + y^{4} + 6 x^{2} y^{2} = c$

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