358 CHAPTER. 9 • Z-TR.ANSFOR.MS AND APPLICATI ONS
9.2 Second-Order Homogeneous Difference Equations
Before proceeding with the z-transform method, we mention a heuristic method
based on substitution of a trial solution. Consider the second-order homogeneous
linear constant-coefficient difference equation (HLCCDE)
y[n + 2] - 2ay[n + 1] + by{n] = 0 (9-8)
where a and b are constants. Using the trial solution y(n] = rn, direct substitu-
tion into (9-8) produces the equation rn+^2 - 2arn+I + bT'" = o. Dividing through
by r" produces the characteristic polynomial r^2 - 2ar + b and characteristic
equation
r^2 - 2ar + b = 0. (9-9)
There are three types of solutions to (9-8), which are determined by the nature
of the roots in (9-9).
Case (i) If b < a^2 , then we have real distinct roots r 1 = a - ~ and
r2=a+~.and
y(n] = c1 rf + c2r~. (9-10)
Case (ii) If b = a^2 , then we have a double real root r = r 1 = r 2 =a, and
(9-11)
Case (iii) If b > a^2 , then we have two complex roots r 1 = a - i./b=(i'i and
r2 =a+i~, and
(9-12)
The solution for case (iii) can also be written as the following linear combination:
(9-13)
where r = VfJ and </> = arctan(-.1°;;"-
2
).
Caution. Be sure to use the value of arctan that lies in the range 0 < </> :$ 1r.