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$Solution of the equation x d y - [y + x y^{3} (1 + \log x)] d x = 0 is$

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a

−x2y2=2x3323+logx+c

b

x2y2=2x3323+logx+c

c

−x2y2=x3323+logx+c

d

none of these

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detailed solution

Correct option is A

We have xdy−ydx=xy3(1+log⁡x)dx⇒−ydx−xdyy2=xy(1+log⁡x)dx Observing the complete differentials, we get −dxy=xy(1+log⁡x)dx⇒−xydxy=x2(1+log⁡x)dx−xy2=(1+log⁡x)x33−∫x33⋅1xdx⇒−x22y2=x33(1+log⁡x)−x39+c2⇒−x2y2=2x3323+log⁡x+c

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