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

Answer

`cy=1-log|y/x|`

Explanation

We are given

`y dx+xlog(y/x)dy-2x dy=0`


On simplifying,

`dy[xlog(y/x)-2x]=-ydx`


`rArr (dy)/(dx)=-y/(xlog(y/x)-2x)=-y/(-x(2-log(y/x)))`


`rArr (dy)/(dx)=-y/(-x(2-log(y/x)))`


`rArr (dy)/(dx)=(y/x)/(2-log(y/x))-(1)`


Now the equation is homogenous as

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