adj. 1. (of a polynomial) having all its terms of the same degree. For example, x² + 2xy + y² is homogeneous in the second degree.
2a. (of an equation) consisting of an equality between a homogeneous function and zero.
b. (of a system of linear equations) having the form Ax = 0. where x is the vector of variables, 0 is the zero vector, and A is the matrix of coefficients. 3. (of an ordinary differential equation) able to be as expressed as of the form y⁽ⁿ⁾ = f(x, y); it is homogeneous of degree γ if
f ( λ x, λ y) = λ^γ f(x, y).
Ordinary differential equations of any order that are homogeneous of degree zero can be solved by writing them in terms of v = y/x. 4. (of a function on a vector space) such that
f(tx₁,..., tx_n) = t f(x₁,..., x_n)
for every non-zero scalar t. More generally, if
f(tx₁,..., tx_n) = t^d f(x₁,..., x_n)
for every non-zero scalar t then f is homogeneous of degree d; if this holds for positive t, then f is said to be positively homogeneous of degree d.