Show that the given differential equation is homogeneous and solve each of them.`y dx+xlog(y/x)dy-2x dy=0`

Explanation

The given differential equation may be written as
`y(dx)/(dy)=2x-xlog(y/x)`
`=2x-x.log{(1/(x//y))}`
`=2x-x.[log1-logx/y]`
`=2x+xlogx/y`.
`therefore (dx)/(dy)=2(x/y)+(x/y)logx/y=f(x/y)`………..(i)
Putting `x=vy` and `(dx)/(dy)=v+y(dv)/(dx)` in (i), we get
`v+y(dv)/(dy)=2v+v(log v)`
`rArr y(dv)/(dx)=v(1+logv)`
`rArr int1/(v(1+logv))dv=int1/ydy rArr int1/tdt=int1/ydy`, where `(1+logv)=t`
`rArr log|t|=log|y|+log|C_(1)| rArr log|1+logv|-log|y|=log|C_(1)|`
`therefore +logx/y=Cy` is the required solution.