Show that the differential equation `(1+e^(x//y))dx+e^(x//y)(1-x/y)dy=0` is homogenous and solve it.

Explanation

The given differential equation may be written as
`(dx)/(dy)=(e^(x//y)(x/y-1))/(1+e^(x//y))`……………………(i)
This is of the form `(dx)/(dy)=f(x/y)`. So, it is homogenous.
Putting `x=vy` and `(dx)/(dy)=v+y(dv)` in (i), we get
`v+y(dv)/(dx)=(e^(v)(v-1))/(1+e^(v))`
`rArr y(dv)/(dx)={(e^(v)(v-1))/(1+e^(v))-v}=-(v+e^(v))/(1+e^(v))`
`rArr int(1+e^(v))/(v+e^(v))dv=-int1/ydy` [on integrating both sides]
`rArr log|v+e^(v)|+log|y|=log|C_(1)|` Where `C_(1)` is an arbitrary constant
`rArr (v+e^(v))y|=log|C_(1)|`
`rArr log|(v+e^(v))|=log|C_(1)|`
`rArr (v+e^(v))y=+-C_(1)=-C` (say)
`rArr (x/y+e^(x//y))y=C`
`rArr x+ye^(x/y)=C`, which is the required solution.