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11.1 Introduction to Difference Equations I

11.1 Introduction to Difference Equations I. Difference Equation Three-Step Procedure Using Technology. Difference Equation. An equation of the form y n = ay n - 1 + b where a, b and y 0 are specified real numbers is called a difference equation.

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11.1 Introduction to Difference Equations I

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  1. 11.1 Introduction to Difference Equations I • Difference Equation • Three-Step Procedure • Using Technology

  2. Difference Equation • 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.

  3. Example Difference Equation 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.

  4. Example Difference Equation (2) 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.

  5. Three-Step Procedure • 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

  6. Example Three-Step Procedure 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.

  7. Example Three-Step Procedure (2) 2. Graph these terms (0,1), (1, 5), (2, 5.8), (3, 5.96) and (4,5.992).

  8. Example Three-Step Procedure (3) 3. Solve the equation. Here a = .2 and b = 4.8. Therefore,

  9. Using Technology • 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.

  10. Example Using Technology Use an Excel spreadsheet to compute the first few terms and the graph of yn = .2yn - 1 + 4.8 with y0 = 1.

  11. Summary Section 11.1 - Part 1 • 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.

  12. Summary Section 11.1 - Part 2 • 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

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