10 Possible Types Of Solutions

10 Possible Types of Solutions 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:

1. 2. 3. 4. 5. 6.

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

a.

b.

c.

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 by choosing

so that the coefficient of

is zero (sub case a). Since no

solutions, the solutions do not change as the

2.

Consider cases when the coefficient of behavior of

as

get larger.

at all. This is accomplished appears in the

changes. Thus the solution is constant.

is positive (sub case b). After that we observe the

3.

Examine cases when the coefficient of behavior of

For case No 1

as

get larger.

, instead of choosing

, we choose the value of

Cases

Exponentially increasing without bound

,

3

Exponentially decreasing without bound

,

4

Constant

,

5

Linearly increasing without bound

,

6

Linearly decreasing without bound

,

7

Constant

,

8

Exponentially decreasing to a bound

,

9

Exponentially increasing to a bound

,

10

Constant

,

11

Oscillating with decreasing amplitude

,

12

Oscillating with decreasing amplitude

,

13

Constant

,

14

Oscillating with constant amplitude

,

15

Oscillating with constant amplitude

,

16

Constant

,

17

Oscillating with increasing amplitude

,

18

(positive, zero and negative).

Type of Solution Constant

,

2

is negative (sub case c). After that we inspect the

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. 3.

Determine

4.

If

, the value of

5.

If

, the value of

and

Using the value of

, determine which of the six cases cover this equation compare to

will determine the sub case

compare to

will determine the sub case