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If $x^{2} y d x - (x^{3} + y^{3}) d y = 0$ , $y (0) = 1$ and $y^{3} \log y = k x^{3}$ then k =

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Correct option is B

⇒dydx=x2yx3+y3=y/x1+(y/x)3V+xdvdx=v1+v3( put y=vx)⇒xdvdx=−v41+v3∫1+v3v4dv=−∫dxx⇒∫v−4+1vdv=−∫1xdx⇒v−3−3+ln⁡v=−ln⁡x+c⇒ln⁡xyx−13v3=cln⁡y−x33y3=cy(0)=1⇒c=0 ∴log⁡y=x33y3

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If x2ydx−x3+y3dy=0, y0=1 and y3logy=k x3 then k =

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If $x^{2} y d x - (x^{3} + y^{3}) d y = 0$ , $y (0) = 1$ and $y^{3} \log y = k x^{3}$ then k =