10 Posible Types of Solutions

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We can classify the solution of difference equation  (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 solution. 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 solving 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

It is possible to divide the value of  into six cases:

The only possibility not covered by these six cases is  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 , the three sub cases will be

This will cover all possibility of initial value . The comparison of  with  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  at all. This is accomplished by choosing  so that the coefficient of  is zero (sub case a). Since no  appears in the solutions, the solutions do not change as the  changes. Thus the solution is constant.
2. Consider cases when the coefficient of  is positive (sub case b). After that we observe the behavior of   as  get larger.
3. Examine cases when the coefficient of  is negative (sub case c). After that we inspect the behavior of   as  get larger.

For case , instead of choosing , we choose the value of (positive, zero and negative).

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

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

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