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$If x^{3} \frac{d y}{d x} = y^{3} + y^{2} \sqrt{y^{2} - x^{2}} and y = 1 when x = 1 then y = 2 when x is$

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c

6/5

d

8/5

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detailed solution

Correct option is D

dydx=y3x3+y2x3xy2x2-1 Put y=Vx→dydx=V+xdvdx→V+xdvdx=v3+v2v2−1xdvdx=v3-v+v2v2−1xdvdx=vv2-1+v2v2−1xdvdx=v2−1vv2−1+v2dvv2−1vv2−1+v2=dxx→∫dxx=∫dvvv2−1 v+v2−1×v-v2−1v-v2−1∫dxx=∫v-v2−1vv2−1dv∫dxx=∫1v2−1dv−∫1vdvlogx+logc=log(v+v2−1)-logv⇒cx=v+v2−1v=y+y2−x2y Put x=y=1⇒c=1⇒put y=2 ⇒x=2+4-x222x=2+4-x2 ⇒2x-22=4-x2⇒(5x−8)x=0→x=85

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