Question

5. Take the difference equation Yn = -1 + 31, Yo = 40 and do the following: A. Generate the first 5 values of the list using the difference equation. Show how you generated the values. Place a box around your final answer. (2 points) B. Solve the difference equation. Show all the steps that you performed to arrive at your answer. Show the general formula that you used. Place a box around your final answer (4 points) C. Use the solution of the difference equation to determine the value for ys (2 points)

Answer 1

a) First few values are:

n	y_n
0	40
1	-24
2	64
3	-57
4	109.375
5	-119.3906

b) we get

Yn = In + B

In +B=- (In-1 +B) +31

So that

11 In + B = -=rn-1 +31 - 11B

We choose B so that

11B B = 31 - 19B -= 31 B= 31 x 8 19

For which we get meaning

= - In-1

$x_1=-\frac{11}{8}x_{0},x_1=\left(-\frac{11}{8} \right )^2x_0$ and so

In general we get $x_n=\left(-\frac{11}{8} \right )^nx_0$ where

31 x 8 512 Yo = 10 + B = 20 = yo- B = 40 - 1919

And so is the required general solution

248 Yn = In + B = yn = 6n + B = 512 (1) 28 512 19

c) Value is

ys 357.35

$\blacksquare$

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