Page 346 - Schaum's Outline of Differential Equations

CHAP. 34] AN INTRODUCTION TO DIFFERENCE EQUATIONS 329

Equation (2) gives us the accrued amount of money after n months. To find the amount of money compiled
after 5 years, we let n = 60 in (2) and find that y 60 = 3623.39.

Supplementary Problems

In Problems 34.17 through 31.20, consider the following difference equations and determine the following: (1) the inde-
pendent variable; (2) the dependent variable; (3) the order; (4) whether they are linear; (5) whether they are homogeneous.

34.17. u a+1=

3418. w k = 6* + k + 1 + In w k_^

34.19. Z ( + Z (+i + Z (+2 + Z (+3 = 0.
34.20. g m_ 2 = 7g m+2 + g m+11

34.21. Verify a n = Ci(2) n + c 2(-2) n satisfies a n+2 = 4a n, where Cj and c 2 are any constants.

34.22. Verify b n = Ci(5)" + c 2w(5)" satisfies b n+2 - Wb n+i + 25b n = 0, where Cj and c 2 are any constants.

34.23. Verify r n = satisfies r n+2 = 6r n+1 — 5r n + 1, subject to r Q = 1, r l = 0.
34.24. Find the general solution to k n+i = —17k n.

34.25. Find the general solution to y n+2 = lly n+i + 12y n.

34.26. Find the general solution to x n+2 = 20x n+1 - W0x n.

n
34.27. Find a particular solution to w n+1 = 4w n + 6" by guessing w n = A(6) , and solving for A.
2
34.28. Find the general solution to v n+1 = 2v n + n .
34.29. Solve the previous problem with the initial condition v 0 = 7.

34.30. Solve Fibonacci's equation/ n+2 =/„+! +/„, subject to/ 0 =/j = 1.

34.31. Suppose you invest $500 on the last day of the month at an annual rate of 12%, compounded monthly. If you invest
an additional $75 on the last day of each succeeding month, how much money would have been accrued after ten
years.