The polynomial function, where the degree of each term is *n *and

f(x,y) = a_{0}x^{n} + a_{1} x^{n-1 }+ a_{2}x^{n-2}y^{2} +…….+ a_{n}y^{n}

is called a homogeneous function of degree *n *x and y. Again.

f(x,y) = a_{0}x^{n} + a_{1} x^{n-1 }+ a_{2}x^{n-2}y^{2} +…….+ a_{n}y^{n}

= x^{n} [a_{0} + a_{1} (y/x) + a_{2} (y/x)^{2} +…..+ + a_{n}(y/x)^{n}]

= x^{n}f(y/x)

Thus, a function f(x,y) is said to be homogeneous function of degree *n *in x and y, if

f(x,y) = x^{n} f(y/x)

**Definition: **

A function z = f(x,y) is said to be homogeneous of degree* n (n *being a constant) if, for any

real number λ

F(λ× , λy ) = λ^{n} f(x,y)

Thus, if both x and y are multiplied by the same real number, then the resulting function value is a power of the number times the function value f(x,y).

**Example : If **f(x, y) = 2x^{2}y+ xy^{2} – y^{3}

F(λ× , λy ) = 2( λx)^{2 }( λx) + (λx) (λy)^{2 }-(λy)^{3}

=2λ³x^{2}y+ λ³xy^{2 }- λ³y^{3}^{ }

= λ³f(x,y)^{ }