Show that the differential equation (x e^y/x + y)dx = xdy is homogeneous. Find the particular solution of this differential equation, given that x = 1 when y = 1.

Given differential equation,

….(1)

Clearly, it is an homogeneous differential equation.

Now, let …(2)

…..(3)

Putting (2) and (3) in (1),

Integrating both the sides, we get

Put

For particular solution put

So, the particular solution of given differential equation is=

∵ log e =1

⇒y=x - x log(1- e log x)