What is Homogeneous? What is homogeneous?

"Qi times" literally means "equal times".

There are two places in differential equations where the term "homogeneous" is used:

1. Shaped like y'The equation of =f(y x) is called a "homogeneous equation", which means that the degree of each term in the equation with respect to x and y is equal, for example, x 2, xy, y 2 are all quadratic terms, and y x is considered to be 0 terms, and the equation y'Each term in =1+y x is a 0th order, so it is a "homogeneous equation".

2. Shaped like y''+py'The equation with +qy=0 is called a "homogeneous linear equation", where "homogeneous" refers to each term in the equation with respect to the unknown function y and its derivative y',y'',……are all equal (all once), and Eq. y''+py'+qy=x is not "homogeneous", because the term x on the right side of the equation does not contain the derivatives of y and y, and is about y, y',y'',……, so it is called a "non-homogeneous linear equation".

In addition, there is also a name for "homogeneity" in sexual algebra, for example, f=ax 2+bxy+cy 2 is called quadratic homogeneity, that is, the meaning of quadratic homogeneity, because each term in f is about the quadratic terms of x and y. Homogeneous polynomial :

qí cì duō xiàng shì

"Qi times" literally means "equal times". When the same kind of terms are combined, the polynomial has the same order.

For example, x *y and z are homogeneous; 3x*x y*y *z*z x*y y*z is a quadratic homogeneous formula.

For example, the square of x plus 2 times xy plus 3 times the square of y, so that the quadratic term here means that the degree of each term in the equation with respect to x and y is equal, so it is a quadratic homogeneous formula, and the homogeneous polynomial is similar.

Sine and cosine homogeneous expressions refer to expressions.

Medium, sine and cosine functions.

is the same for the index.

For example: tanx=2, find: (sinx+3cosx) (sinx-4cosx).

The above equation is the homogeneous formula of sinx and cosx, which can be found by turning it into tanx.

Numerator denominator. Same as dividing by cosx, then, the original formula = (tanx+3) (tanx-4) = -5 2.

Turning homogeneous expressions of sin and cos into expressions of tgx is a common technique that should be mastered proficiently.

The so-called homogeneity is the meaning of qi, and after merging the same terms, all the times are the same.

For example, x 2y, 3z is a homogeneous formula;

For example, the square of x plus 2 times the square of xy and 3 times the square of y, so that all the quadratic terms are aligned, so it is a quadratic homogeneous formula, and the homogeneous polynomial is similar.

Qi, as the name suggests, is neat and consistent, because only the polynomial degree is the same.

Recognize what homogeneous forms are.

Homogeneous is a special kind of multivariate polynomial. IfNumber of fieldsIf the degree of each item of the n-yuan polynomial on p is equal to m, then the polynomial is said to be n-yuan mHomogeneous polynomial :, referred to as m-order homogeneous, also known as m-order type of n variables.

The primary form is also called the linear type, the product of two n-yuan homogeneous polynomials is still a homogeneous polynomial, and the order is equal to the sum of the degrees of these two homogeneous polynomials, and any n-ary polynomial on the number field p can be uniquely expressed as the sum of the homogeneous polynomials on p.

Decomposition theorem:

Any polynomial in f[x] of no less than 1 can be decomposed into the product of the irreducible polynomial on f, and the factor is removed.

In addition to the order and constant factor, the method of decomposition is unique.

When f is a complex number field c, according to the fundamental theorem of algebra.

The irreducible polynomials in provable c[x] are all once. Thus, each polynomial of complex coefficients can be decomposed into a continuous product of a primary factor.

When f is the field of real numbers r, the irreducible polynomial in r[x] is once or twice the polynomial since the imaginary roots of the real state trillion polynomials appear in pairs, i.e., the common congregation of the imaginary roots is still the root. So each real coefficient polynomial can be decomposed into the product of some irreducible polynomials of primary and quadratic. A sufficient and necessary condition for the quadratic polynomial of the real coefficient x2+bx+ irreducible.

is its discriminant B2-4 <0.

A homogeneous formula means that the numerator and denominator of a trigonometric formula are of the same order.

Such as sinx cosx......Note that the chain meaning (sinx) 2 is also a homogeneous answer, because the denominator 1 can be rewritten as (sinx) 2+(cosx) 2

The general treatment method is to multiply the trigonometric function by the trigonometric function of the same division up and down of the homogeneous formula Bu Kai, so as to achieve the purpose of string tangent and turn the heteronymous trigonometric function into the trigonometric function of the same name.

Recognize the history of the reed, and recognize the uproar as the Qi Shi state.

"Homogeneous" refers to the combination of dust tremors and similar terms.

, each term is an equal polynomial with respect to x, y.

The number of times is a homogeneous formula, and the number of times is a quadratic homogeneous formula, such as: x 2y, 3z is a homogeneous formula, and x 2+xy is a quadratic homogeneous formula.

Homogeneous equations. Homogeneousequation) is an equation in mathematics, which refers to the exponential equality of all non-zero terms in the simplified equation, also known as the number of unknowns about the remaining terms of each defeated brother. The left end of the equation is the term with unknowns, and the right end is equal to zero.

In general, the homogeneous equation is a transitional form for solving the problem, which is easy to solve after being converted into a homogeneous equation.

A polynomial.

The equation in which the order of each monomial is the same is called homogeneous.

Qi Ci" literally means that "the number of secondary postures is equal".

For example, x+y+z is 1 times

Or x 2+2x*y+y 2 times are all 2

Or x 3+x*y*z+y 3+z 3 times are all 3 times to tease these 3 are all homogeneous.

"Homogeneous fraction" means that in a fraction, all but a finite number of zeros are finite. This concept was proposed by Cauchy, who mentioned it in his Analytics. In a fraction, if all the zeros are finite, then the fraction is homogeneous.

If all the zeros in a fraction are finite, then the fraction is a "homogeneous fraction". There is a special case of homogeneous fractions, that is, the number of "zeros" in a fraction is finite, and this fraction is also homogeneous.

That is, the numerator and denominator are all of the same order, for example.

x 2+xy) (y 2+yz) cannot appear as a single term and a constant term.

1. The homogeneous formula refers to the polynomial that after merging similar terms, the number of times of each land closed argument item about x and y is equal, the number of times is a homogeneous formula, and the number of times is a homogeneous formula, such as x 2y, 3z is a homogeneous formula, and x 2+xy is a quadratic homogeneous formula. A homogeneous equation is an equation in mathematics, which refers to the exponential equality of all non-zero terms in a simplified equation, also known as the number of unknowns of each item contained.

2. The left end of the equation is the term with unknowns, and the right end is equal to zero. In general, the homogeneous equation is a transitional form of solving the problem, which is easy to solve after being converted into a homogeneous equation.

Explanation of Qi Ci.

homogeneous]

In an algebraic equation or bend, all terms are of the same order.

Word decomposition Explanation of Qi Qi (Qi) í One end of something is flat or arranged in a straight line: neat. Uneven.

Shirt bucket boring reached, with what is the same as the same: see the sage and think together. The river is waist-deep.

Meantime; Equally; Together: the same name. Chorus.

Together. Let's move forward together. Whole; Completely:

Complete. The people are here. Interpretation of the Chinese Zhou Dynasty Princes and States Times Ì Second:

Next. Second son. Inferior.

Minor. Quality, poor quality: defective products.

Defective goods. Ranking, order: order.

Order. Position. Chemically refers to acid groups or compounds that contain less than two oxygen atoms

Hypochlorous acid. Middle: Chest.

Measure word, back: number of times (?For the first time.

Three pins are clear.

A polynomial in which all the positive mials have the same degree is called homogeneous. For example, change late:

x+3y+z

Or grinding the penitent x 2+x*y+y 2 is the homogeneous formula. [