Download Difference Equation

1.
2.
3.
Difference Equation
Three-Step Procedure
Using Technology
1



An equation of the form
 yn = ayn - 1 + b
where a, b and y0 are specified real numbers is
called a difference equation.
The starting value, y0, is called the initial value.
2
Suppose a savings account contains $40 and
earns 6% interest, compounded annually. At the
end of each year a $3 withdrawal is made.
Determine the difference equation that describes
how to compute each year's balance from the
previous year's balance.
3
Balance
+ Interest for year
- Withdrawal
(.06)40
$3
y1 = 39.40
(.06)39.40
3
y2 = 38.76
(.06)38.76
3
y0 = $40
The balance for the next year equals the previous
year's balance plus the interest earned on the
previous year's balance minus the withdrawal.
yn = yn - 1 + .06yn - 1 - 3 = 1.06yn - 1 - 3.
4




For the difference equation yn = ayn - 1 + b:
1. Generate the first few terms.
2. Graph the terms.
Plot the points (n, yn) for n = 0, 1, 2, …
3. Solve the difference equation. The solution is
b
b  n

yn 
  y0 
 a , a  1.
1 a 
1 a 
5
Study the difference equation yn = .2yn - 1 + 4.8
with y0 = 1.
1. Generate the first few terms.
y0 = 1
y1 = .2(1) + 4.8 = 5
y2 = .2(5) + 4.8 = 5.8
y3 = .2(5.8) + 4.8 = 5.96
y4 = .2(5.96) + 4.8 = 5.992.
6
2. Graph these terms (0,1), (1, 5), (2, 5.8),
(3, 5.96) and (4,5.992).
7
3. Solve the equation.
Here a = .2 and b = 4.8.
Therefore,
4.8 
4.8 
n
yn 
 1 
 .2 
1  .2  1  .2 
 6  5 .2  .
n
8





An Excel spreadsheet can be used to evaluate
and graph the first few terms of a difference
equation.
1. Enter 0 into cell A2, enter 1 in cell A3, select
the two cells, and drag the fill handle down to
A8.
2. Enter the value for y0 into cell B2.
3. Enter the formula for yn into cell B3 using B2
in place of yn - 1, select cell B2 and B3 and drag
the fill handle down to B8.
4. Highlight cells A2:B8 and use the Chart
Wizard to make an XY(Scatter) type chart.
9
Use an Excel spreadsheet to compute the first few
terms and the graph of yn = .2yn - 1 + 4.8 with y0 = 1.
10

A difference equation is an equation of the form
yn = ayn - 1 + b, where a, b, and y0 are specified,
and determines a sequence of numbers in
which each of the numbers (that is, y0, y1,…) is
obtained from the preceding number by
multiplying the preceding number by a and
adding b. The first number in the sequence, y0,
is called the initial value.
11


The graph of a difference equation is obtained
by graphing the points (0, y0), (1, y1), (2, y2), …
The value of the nth term of a difference
equation yn = ayn - 1 + b (y0 given and a 1) can

be obtained directly (that is, without generating
the preceding terms) with the formula
b
b  n

yn 
  y0 
a .
1 a 
1 a 
12