How to explain the concept of difference equations, difference equation solving formulas

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.

Difference equations. It refers to an equation that contains the difference and the independent variables of an unknown function, and then finds the filial piety quietly like a differential equation.

The numerical solution of , in which the differentiation is often approximated by the contrast difference, and the derived equation is the difference equation. Finding an approximate solution to a differential equation by solving a split equation is an example of the discretization of the lead transport continuity problem.

Methods for solving equations. First, observe the equation, second, use the properties of the equation to solve the equation, and third, merge similar terms to deform the equation into a monomial.

Fourth, move terms to move the term with unknown numbers to the left, and the constant term to the right five. Remove the brackets, use the bracket rule, remove the brackets in the equation, and solve the four rules of Qiaoqi.

When finding the numerical solution of a differential equation, the differential is often approximated by the contrast difference, and the derived 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*.

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.