The degree of 1. equation: first, to judge whether an equation is homogeneous, it is necessary to ensure that the sum of the degrees of the equation is equal. The degree of polynomial equation refers to the degree of the highest power of each term in the equation. If the degree of the equation is equal, then the first condition of the homogeneous equation is satisfied.
The sum of the degrees of this equation is 5, because $2+3=5$ is a homogeneous equation.
2. Constant term: another characteristic of homogeneous equation is that it does not contain constant term, that is, the constant coefficients in the equation are all 0. If there are constant terms in the equation, it is not a homogeneous equation.
This equation contains constant term 4, so it is not homogeneous.
3. Homogeneity: Homogeneous equation has an important property, that is, if a polynomial equation is homogeneous, then every term of it can be divisible by a common non-zero factor. This means that if you multiply all the variables in the equation by the same non-zero constant at the same time, the equation still holds.
For example, consider the homogeneous equation $ 2x 2y 3-4x 3y 2 = 0 $. If $x$ and $y$ in the equation are multiplied by 2, it is obtained that $ 4x 2y 3-8x 3y 2 = 0 $ is still a homogeneous equation.
4. General form of homogeneous equation: The general form of homogeneous equation can be expressed as:
F(x，y)=0
Where $F(x, y)$ is a polynomial function, which satisfies the condition of homogeneous equation, that is, the sum of degrees is equal and there is no constant term.
5. Application of homogeneous equation: Homogeneous equation is widely used in mathematics and physics. In calculus, homogeneous equations are often used to solve differential equations; In physics, they can be used to describe many natural phenomena, such as electric field, magnetic field and fluctuation.
In short, to judge whether an equation is homogeneous, we need to consider its degree, constant term and homogeneity. Homogeneous equation is a special kind of equation with some unique properties and applications, which is very helpful to understand the problems in mathematics and physics.