A function f(x,y) is said to be homogeneous function in which the power of each term is same.
A function f(x,y) is a homogeneous function of order n, if the degree of each of its terms in x and y is equal to n. Thus
a_{0} x^{n} + a_{1}x^{n1}y + a_{2}x^{n2}y^{2} + ... + a_{n1}xy^{n1} + a_{n}y^{n} is a homogeneous function of order n.
x^{n}[a_{0} + a_{1}(y/x) + a_{2}(y/x)^{2} + ... + a_{n1}(y/x)^{n1} + a_{n}(y/x)^{n}] = x^{n}φ(y/x)
Euler's Theorem for homogeneous Equations
 If z is a homogeneous function of x,y of order n, then
x ∂z/∂x + y ∂z/∂y = nz
Deductions from Euler's equation
 If z is a homogeneous function of x,y of order n, and z = f(u), then
x ∂u/∂x + y ∂u/∂y = nf(u)/f '(u)
 If z is a homogeneous function of x,y of order n, and z = f(u), then
x^{2} ∂^{2}u/∂x^{2} + 2xy ∂^{2}u/∂x∂y + y^{2} ∂^{2}u/∂y^{2} = g(u)[g'(u)  1]
where g(u) = n f(u)/f'(u)
