Examples of difference equations, three basic methods for solving difference equations

Solve difference equations.

The three basic methods are the classical solution, the recursive solution, and the transformation method.

A fractional equation, also known as a recursive relation, is an equation that contains an unknown function and its difference, but no derivative. The function that satisfies this equation is called the solution of the difference equation. Difference equations are differential equations.

discretization. Difference equation by Huaizhou.

Difference equations of the kth order about the series:

xn-a1xn-1-a2xn-2-……akxn-k=b (n=k,k+1，……

where a1, a2,--ak are constant first masks, ak≠0If b=0, then the square Ming Qingcheng is a homogeneous equation.

Algebraic equations about .

k-a1 k-1---ak-1 -ak=0 is the corresponding eigenequation, and the root is the eigenvalue.

The difference equation solves the Gregorian calendar split: first find the general solution of the homogeneous order, and then find the special solution of the non-homogeneous order, which together is the general solution.

Difference equations include the difference of unknown functions and the equations of independent variables. When finding the numerical solution of a differential equation*, the differentiation is often approximated by the corresponding difference, and the resulting equation is the difference equation. Finding an approximate solution to a differential equation by solving a difference equation is an example of discretization of a continuous problem*.

Mathematically, the recurrence relation, also known as the difference equation, is a family of equations that recursively define a sequence: each item of the sequence is a function of the previous term. Some simply defined recursive relations can exhibit very complex (chaotic) properties, and they belong to the field of nonlinear analysis in mathematics.

Theorem 1 (superposition principle of homogeneous linear difference equation solutions).

If y1(t), y2(t) ,...ym(t) is the homogeneous linear difference equation yt+n+a1yt+n-1 +a2yt+n-2+....m special solutions (m2) of +an-1yt+1+anyt=0, then its linear combination y(t)=a1y1(t)+a2y2(t)+....amym(t) is also the solution of the equation, where a1, a2 ,..., am is an arbitrary constant.

Theorem 2n order homogeneous linear difference equation yt+n+a1yt+n-1 +a2yt+n-2 +....an-1yt+1+anyt=0 must have n linearly independent special solutions.

Theorem 3 (homogeneous linear difference equation general solution structure theorem).

If y1(t), y2(t) ,...yn(t) is the homogeneous linear difference equation yt+n+a1yt+n-1 +a2yt+n-2 +....n linearly independent special solutions of an-1yt+1+anyt=0, then the general solution of the equation is: ya(t)=a1y1(t)+a2y2(t)+....anyn(t), where a1, a2 ,..., and an is n arbitrary (independent) constants.

The square of the difference equation y is 2 1=2 +2a+b+1 =73 (-2)=3 +3a-2b+(-2) =23.

It is widely used in problem solving, and the punch wheel involves the value range of the solution coefficient, the number and distribution of the root of the equation, and the pure condition of the judgment. The discriminant formula for the root of the unary quadratic equation ax 2+bx+c=0(a≠0) is b 2-4ac, denoted by " " (pronounced "delta").

Difference equation: Let it be a real sequence if the following relationship ut-1ut-1 -... is satisfied-put-p=h(t), where 1, 2...., p is the real number and h(t) is the known real function of t, then the above equation is called the linear difference equation satisfied.

If the deterministic function ut,h (t) in the above equation is replaced by a random sequence with known statistical properties, then the linear random difference equation is obtained. Such a wide range of models are not discussed in time series analysis.

xt-ᵠ1xt-1-…-pxt-p=εt-θ1εt-1-…-q t-g where 1, ....p, and 1, ....g is a real number, is a zero-mean stationary series, is a stationary white noise sequence, and when s>t, e sxt=0, the specific linear random difference equation mentioned above is the arma (p,g) model in time series analysis.

First seek the general solution of the same order, and then seek the special solution of the non-uniform order, and the sum is the general solution.

The right side of the homogeneous solution equal sign is 0, that is, f(x+1)-(f(x))=0 The general solution can be obtained according to the formula f(x)=c(-1) x

The solution of non-homogeneous order adopts the general method. For a difference equation of the form F(T+1)-af(t)=CB T, if A is not equal to B, the special solution can be changed to F*(T)=KB T

Substituting the original formula yields kb (t+1)-akb t=cb t to obtain k=c (b-a).

i.e. y=(cb t) (b-a).

The question you gave was a=-1, b=2, c=1

The special solution of the nuclear trap is (2 t) 3 with f(x).

Therefore, the general solution of f(x) is (2 t) 3+c(-1) x c is a slag real number.