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

Explanation

`ydx+x log((y)/(x)dy-2xdy=0`
`impliesydx+[xlog((y)/(x))-2x]dy=0`
`implies(y)/(x)+[log((y)/(x)-2](dy)/(dx)=0`
`implies (dy)/(dx)=[((y)/(x))/(2-log((y)/(x)))]`..........`(1)`
Let `y=vx` (`:'` equation is homogenous)
`implies (dy)/(dx)=v+x(dv)/(dx)`
put in equation `(1)`,
`v+x(dv)/(dx)=(v)/(2-logv)`
`implies x(dv)/(dx)=(v)/(2-logv)-v`
`=(vlogv-v)/(2-logv)`
`implies (2-logv)/(v(logv-1))dv=(dx)/(x)`
`implies int(2-logv)/(v(logv-1))dv=int(dx)/(x)`
Let `logv=t
``implies (1)/(v)dv=dt`
`implies int(2-t)/(t-1)dt=int(dx)/(x)`
`implies int(-1+(1)/(t-1))dt=int(dx)/(x)`
`implies-t+log(t-1)=logx+logc`
`implies -logv+log(logv-1)=log(cx)`
`implies (logv-1)/(v)=cx`
`implies logv-1=cxv`
`implies log((y)/(x))-1=cy`