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Show that the given differential equati...

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`

Text Solution

Verified by Experts

`Use x/y =v`
`x=v⋅y dx=v dy+dv⋅y`
`(1+e^v)(v dy+dv⋅y)+e^v (1−v)dy=0`
On simplification, we get
`dv(y)(1+e^v)+dy(v+e^v)=0`
`⇒ log∣v+e^v∣=−log∣y∣+logc`
`⇒log x/y+e^x/y ∣y∣=logc`
`⇒ ( x/y+e^x/y)(y) =c`
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