Show that the given differential equation is homogeneous and solve each of them.`(1+e^(x/y))dx+e^(x/y)(1-x/y)dy=0`

Explanation

`(1+e^(x//y))dx+e^(x//y)(1-(x)/(y))dy=0`
`implies (1+e^(x//y))dx=-e^(x//y)(1-(x)/(y))dy`
`implies (dx)/(x)=(-e^(x//y)(1-(x)/(y)))/(1+e^(x//y))`...........`(1)`
It is a homogenous differential equation.
Let `x=vy`
`implies (dy)/(dx)=v+y.(dv)/(dy)`
put in equation `(1)`,
`v+y.(dv)/(dy)=(-e^(v)(1-v))/(1+e^(v))`
`implies y(dv)/(dy)=(-e^(v)+ve^(v))/(1+e^(v))-v`
`implies y.(dv)/(dy)=(-e^(v)+ve^(v)-v-ve^(v))/(1+e^(v))`
`=(-(e^(v)+v))/(1+e^(v))`
`implies(1+e^(v))/(v+e^(v))dv=-(dy)/(y)`
`implies int(1+e^(v))/(v+e^(v))dv=-int(dy)/(y)`
`implies log(v+e^(v))=-logy+logc`
`implies log(v+e^(v))+logy=logc`
`implieslog(vy+ye^(v))=logc`
`impliesvy+ye^(v)=c`
`impliesx+y*e^(x//y)=c`