Homogeneous Equation
What is a homogeneous equation?
An equation is called a homogeneous equation if it consists of a homogeneous polynomial set equal to zero: $$ P(x)=0 $$
What is a homogeneous polynomial?
A homogeneous polynomial is a polynomial in which all the terms (monomials) have the same total degree.
How is the degree of a monomial determined? The degree of a monomial is obtained by summing the exponents of its variables. For example, xy2 is a monomial of degree three, because the exponent of x is 1 and that of y is 2, so their sum is 1+2=3.
Similarly, a system of equations that consists exclusively of homogeneous equations is called a homogeneous system of equations.
A practical example
Example 1
This equation is homogeneous:
$$ 2xy^2+4x^3-4y^3 = 0 $$
because every term is of degree three.
Example 2
This equation is homogeneous:
$$ 2x^2+4xy+3y^2 = 0 $$
because all its terms are of degree two.
Example 3
This equation is NOT homogeneous:
$$ 3x+5xy+7y^2 = 0 $$
because the term 3x has degree one, while the other two terms have degree two.
Example 4
This equation is NOT homogeneous:
$$ 3x^2+5xy+7y^2 = 5 $$
Although the polynomial P(x)=3x2+5xy+7y2 is homogeneous (since all its terms have the same degree), the equation itself is not homogeneous because the homogeneous polynomial is not equal to zero, that is, P(x)≠0.
And so forth.
