Question

# Solution of the differential equation √xdx+√ydy√xdx−√ydy=√y3x3 is given by

A

32log(yx)+logx3/2+y3/2x3/2+tan1(yx)3/2+c=0

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B

23log(yx)+logx3/2+y3/2x3/2+tan1yx+c=0

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C

23log(yx)+log(x+yx)+tan1(y3/2x3/2)+c=0

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D

None of the above

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Solution

## The correct option is D None of the above We have, √xdx+√ydy√xdx−√ydy=√y3x3 ⇒d(x3/2)+d(y3/2)d(x3/2)−d(y3/2)=y3/2x3/2 ⇒du+dvdu−dv=vu, where u=x3/2 and v=y3/2 ⇒u du+u dv=v du−v dv ⇒u dv+v dv=v du−u dv ⇒u du+v dvu2+v2=v du−u dvu2+v2 ⇒d(u2+v2)u2+v2=−2d(tan−1(uv)) On integrating, we get log(u2+v2)=−2 tan−1(uv)+c ⇒12log(x3+y3)+tan−1(xy)3/2=c2=C Where, c and C are constants.

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