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Question

Solution of the given D.E y2dx+(x2xy+y2)dy=0 is

A
tan1(yx)logy+c=0
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B
2tan1(yx)logx+c=0
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C
log(y+x2+y2)+logy+c=0
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D
log(yx2+y2)logy+c=0
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Solution

The correct option is A tan1(yx)logy+c=0
y2dx+(x2xy+y2)dy=0(x2xy+y2)dydx+y2=0
Substituting y=xvdydx=v+xdvdx
(x2x(xv)+x2v2)(v+xdvdx)+x2v=0x2(xdvdx+v3+xv2dvdx+vxvdvdx)=0dvdx=v3vx(v2v+1)dvdx×v2v+1v3v=1x
Integrating both sides, we get
dv(v2v+1v3v)=1xdxtan1vlogv=logx+ctan1vlogvlogx+c=0tan1yxlogy+c=0

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