How to solve a difference equation and find a general solution to a difference equation

Mathematically, a recurrence relation, or difference equation.

difference equation), which is an equation that recursively defines a sequence.

Each item of the sequence is a function defined as 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.

The so-called solution of a recursive relation, that is, the analytical solution of it, that is, a non-recursive function with respect to n.

in numerical analysis.

The first problem encountered in this is how to put differential equations.

The solution of the difference equation can best approximate the solution of the original differential equation, followed by the calculation.

For example, dy+y*dx=0,y(0)=1 is a differential equation, and x takes the value [0,1].

Note: The solution is y(x)=e (-x));

To discretize a differential equation, the interval of x can be divided into many small intervals [0,1 n],[1 n,2 n],n-1)/n,1]

In this way, the above differential equation can be discretized as: y((k+1) n)-y(k n)+y(k n)*(1 n)=0, k=0,1,2,..n-1 (n systems of discrete equations).

Using the condition y(0)=1 and the difference equation above, we can calculate an approximate value of y(k n).

Then find a special solution, set a special solution according to the form of excitation, and then substitute it into the original difference equation.

The undetermined coefficients that can be found in the special solution are obtained, and the sum of the homogeneous solution and the special solution is the full response, and the initial value is used to find the undetermined coefficient after the form of the full response is set, such as the initial values y(0) and y(1) are used in the second-order system.

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 the discretization of a continuous problem.