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Question

Solve the differential equation:
x2dydx=x22y2+xy.

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Solution

Given,
x2dydx=x22y2+xy
dydx=12y2x2+yx
substitute y=vxdydx=xdvdx+v
xdvdx+v=12x2v2x2+xvx
xdvdx+v=12v2+v
xdvdx=12v2
dv12v2=dxx
integrating on both sides, we get,
dv12v2=dxx
122log1+2v12v=logx+c
122logx+2yx2y=logx+c

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