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Solution of Affine Difference Equation
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 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 _{ }
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:
 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.
 Consider cases when the coefficient of _{ } is positive (sub case b). After that we observe the behavior of _{ } as _{ } get larger.
 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).
No 
Cases 
Type of Solution 
1 
_{ } , _{ } 
Constant 
2 
_{ } , _{ } 
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 
_{ } , _{ } 
Oscillating with increasing amplitude 
Step by step to determine the solution of first order linear difference equation
 Put the equation into form of Equation (1 ).
 Determine _{ } and _{ }
 Using the value of _{ } , determine which of the six cases cover this equation
 If _{ } , the value of _{ } compare to _{ } will determine the sub case
 If _{ } , the value of _{ } compare to _{ } 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\