These are concrete types, including first-order linearity, first-order nonlinearity, second-order linearity and so on.
Let's take the common first-order linear differential equation as an example.
The standard form of the first order linear differential equation is
dy/dx + yP(x) = Q(x)
The differential equations in the above formulas are all called first-order linear differential equations, and vice versa.
If Q(x)=0, the above equation is called a first-order linear homogeneous differential equation, and vice versa.
For example:
dy/dx = y + x ^ 2
Dy/dt = x * symplectic+t 2
Are linear differential equations that conform to the above formula.
y * y' -2*xy = 3
y' - Cosy = 1
They do not conform to the standard form of first-order linear differential equations, so they are not.
Standard form of Bernoulli equation
dy/dx + P(x)*y = Q(x) * y ^ n
Anything that conforms to the above form is called Bernoulli equation.
Read the definition in the textbook carefully, don't look at many examples, just distinguish my definition.
Explain clearly and easily! ! ! !
In the case of linear algebra, order refers to the number of rows of determinant. Because determinant is a set of numbers.
1 2 3 4
2 3 4 5
1 2 1 1
4 3 2 5
Enclose it in braces. The determinant above has four rows and four columns, so it is called a fourth-order determinant.
8 3 3
1 6 4
4 9 0
Three rows * three columns enclosed in brackets are called third-order determinants.
That is to say, the number of determinant rows = the number of columns = the order. Understand?
Linear relationship is embodied in matrix and space. There is a mathematical relationship between them. Reflected in a certain number of spatial relations between them, it can be expressed by a mathematical expression or space vector.
Linearity can also refer to linear operations, such as:
5A + 43B - 4C + 2 1F = N + F - E
The above formula only contains multiplication and addition and subtraction of numbers, so it is called linear expression, and its operation can be called linear operation.
If it includes operations other than divisor multiplication and addition and subtraction, it can't be a linear operation!