Solve the homogeneous equations. First put the equation in the form of a homogeneous equation.

(x + y) dy + (x - y) dx = 0

A first-order differential equation of the form dy dx is called homogeneous. It can be transformed into an equation whose variables are separable by defining the new variable v = y/x. Then, y = vx and dy dx dx du + = v + x Substitution into the original differential equation and collecting terms with like variables then gives the separable equation dx dv v - F(v) 0. After solving this separable equation, the solution of the original equation is obtained when we replace v by y/x.