The solution of  the differential equation (x² +xy)dy = (x² + y²)dx is-

(x-y)² = kxe^(-y/x)

 The given  differential equation (x² +xy)dy = (x² + y²)dx is

which can be written as dy/dx = (x² +y²)/ (x² +xy).....................(i)

This is an homogeneous differential equation.

Let y = vx

⇒dv/dx = v + x (dv/dx)

substituting the values of v and dv/dx in equation (i) we get:

v+ x (dv/dx) = {x² +(vx)²}/{x² + x (vx)}

⇒x(dv/dx) = (1-v)/(1+v)

⇒{2/(1-v)-1}dv = dx/x

on integrating both sides, we get:

v = - 2 log (1-v)-log x+ log k

since v= y/x

⇒ = (1-v)/(1+v)

Integrating both sides 

(x-y)² = kxe^(-y/x)