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Solution of Affine Difference Equation

We can classify the solution of difference equation Difference Equation Tutorial: 10 possible solutions (linear first order difference equation with constant input) into 10 types and any such difference equation must have a solution that is one of these ten types. There is never more than one type of solutions. By merely examining a difference equation, we will be able to decide which type of solution it has. Moreover, we can make such a statement without the necessity to solve the equation.

The ten possible types of solutions are sorted in the next table .

How does it come out into only 10 possible type of solution? Here is the explanation.

Consider affine dynamical system Difference Equation Tutorial: 10 possible solutions

It is possible to divide the value of Difference Equation Tutorial: 10 possible solutions into six cases:

  1. Difference Equation Tutorial: 10 possible solutions
  2. Difference Equation Tutorial: 10 possible solutions
  3. Difference Equation Tutorial: 10 possible solutions
  4. Difference Equation Tutorial: 10 possible solutions
  5. Difference Equation Tutorial: 10 possible solutions
  6. Difference Equation Tutorial: 10 possible solutions

The only possibility not covered by these six cases is Difference Equation Tutorial: 10 possible solutions that is not allowed in any case if Equation (1) is to be a difference equation. Each of these six cases will be divided into three sub cases. Except for Difference Equation Tutorial: 10 possible solutions , the three sub cases will be

  1. Difference Equation Tutorial: 10 possible solutions
  2. Difference Equation Tutorial: 10 possible solutions
  3. Difference Equation Tutorial: 10 possible solutions

This will cover all possibility of initial value Difference Equation Tutorial: 10 possible solutions . The comparison of Difference Equation Tutorial: 10 possible solutions with Difference Equation Tutorial: 10 possible solutions comes from the value in the parenthesis of the solution in Equation(2) Though it has 6 by 3 = 18 possibilities, there are only 10 possible behaviors of the solutions because some of them are overlapped. The strategy to find the identical solution is as follow:

  1. Look at the solutions given in Equation (2) that do not involve Difference Equation Tutorial: 10 possible solutions at all. This is accomplished by choosing Difference Equation Tutorial: 10 possible solutions so that the coefficient of Difference Equation Tutorial: 10 possible solutions is zero (sub case a). Since no Difference Equation Tutorial: 10 possible solutions appears in the solutions, the solutions do not change as the Difference Equation Tutorial: 10 possible solutions changes. Thus the solution is constant.
  2. Consider cases when the coefficient of Difference Equation Tutorial: 10 possible solutions is positive (sub case b). After that we observe the behavior of Difference Equation Tutorial: 10 possible solutions as Difference Equation Tutorial: 10 possible solutions get larger.
  3. Examine cases when the coefficient of Difference Equation Tutorial: 10 possible solutions is negative (sub case c). After that we inspect the behavior of Difference Equation Tutorial: 10 possible solutions as Difference Equation Tutorial: 10 possible solutions get larger.

For case Difference Equation Tutorial: 10 possible solutions , instead of choosing Difference Equation Tutorial: 10 possible solutions , we choose the value of Difference Equation Tutorial: 10 possible solutions (positive, zero and negative).

No

Cases

Type of Solution

1

* , Difference Equation Tutorial: 10 possible solutions

Constant

2

* , Difference Equation Tutorial: 10 possible solutions

Exponentially increasing without bound

3

* , Difference Equation Tutorial: 10 possible solutions

Exponentially decreasing without bound

4

* , Difference Equation Tutorial: 10 possible solutions

Constant

5

Difference Equation Tutorial: 10 possible solutions , Difference Equation Tutorial: 10 possible solutions

Linearly increasing without bound

6

Difference Equation Tutorial: 10 possible solutions , Difference Equation Tutorial: 10 possible solutions

Linearly decreasing without bound

7

Difference Equation Tutorial: 10 possible solutions , Difference Equation Tutorial: 10 possible solutions

Constant

8

Difference Equation Tutorial: 10 possible solutions , Difference Equation Tutorial: 10 possible solutions

Exponentially decreasing to a bound

9

Difference Equation Tutorial: 10 possible solutions , Difference Equation Tutorial: 10 possible solutions

Exponentially increasing to a bound

10

Difference Equation Tutorial: 10 possible solutions , Difference Equation Tutorial: 10 possible solutions

Constant

11

Difference Equation Tutorial: 10 possible solutions , Difference Equation Tutorial: 10 possible solutions

Oscillating with decreasing amplitude

12

Difference Equation Tutorial: 10 possible solutions , Difference Equation Tutorial: 10 possible solutions

Oscillating with decreasing amplitude

13

* , Difference Equation Tutorial: 10 possible solutions

Constant

14

* , Difference Equation Tutorial: 10 possible solutions

Oscillating with constant amplitude

15

* , Difference Equation Tutorial: 10 possible solutions

Oscillating with constant amplitude

16

Difference Equation Tutorial: 10 possible solutions , Difference Equation Tutorial: 10 possible solutions

Constant

17

* , Difference Equation Tutorial: 10 possible solutions

Oscillating with increasing amplitude

18

* , Difference Equation Tutorial: 10 possible solutions

Oscillating with increasing amplitude

Step by step to determine the solution of first order linear difference equation

  1. Put the equation into form of Equation (1 ).
  2. Determine Difference Equation Tutorial: 10 possible solutions and Difference Equation Tutorial: 10 possible solutions
  3. Using the value of Difference Equation Tutorial: 10 possible solutions , determine which of the six cases cover this equation
  4. If Difference Equation Tutorial: 10 possible solutions , the value of Difference Equation Tutorial: 10 possible solutions compare to Difference Equation Tutorial: 10 possible solutions will determine the sub case
  5. If Difference Equation Tutorial: 10 possible solutions , the value of Difference Equation Tutorial: 10 possible solutions compare to Difference Equation Tutorial: 10 possible solutions will determine the sub case

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Preferable reference for this tutorial is

Teknomo, Kardi (2015) Difference Equation Tutorial. http:\\people.revoledu.com\kardi\ tutorial\DifferenceEquation\