Page 343 - Schaum's Outline of Differential Equations

326 AN INTRODUCTION TO DIFFERENCE EQUATIONS [CHAP. 34

SOLUTIONS
Solutions to difference equations are normally labeled as particular or general, depending on whether there
are any associated initial conditions. Solutions are verified by direct substitution (see Problems 34.8 through
34.10). The theory of solutions for difference equations is virtually identical with that for differential equations
(see Chapter 8) and the techniques of "guessing solutions" are likewise reminiscent of the methods employed
for differential equations (see Chapters 9 and 11).
For example, we will guess y n=p" to solve a constant coefficient, homogeneous difference equation.
Substitution of the guess will allow us to solve for p. See, for example, Problems 34.11 and 34.12.
We will also use the method of undetermined coefficients to get a particular solutions for a non-homogeneous
equation. See Problem 34.13.

Solved Problems

In Problems 34.1 through 34.6, consider the following difference equations and determine the following:
the independent variable, the dependent variable, the order, whether they are linear and whether they are
homogeneous.

34.1. y n+3 = 4y n
The independent variable is n, the dependent variable is y. This is a third-order equation because of the
difference between the highest argument minus the lowest argument is (n + 3) - n = 3. It is linear because of the
linearity of both y n+3 and y n. Finally, it is homogeneous because each term contains the dependent variable, y.

34.2. 4+2 = 4+^3-5^5
The independent variable is i, the dependent variable is t. This is a seventh-order equation because the
difference between the highest argument and the lowest argument is 7. It is linear because of the linear appearances
t
of the it and it is non-homogeneous because of the 4, which appears independently of the t t.
34.3. z kz k+1=W
The independent variable is k, the dependent variable is z. This is a first-order equation. It is non-linear
because, even though both i k and z i+1 appear to the first power, they do not appear linearly (any more than sin i k is
linear). It is non-homogeneous because of the solitary 10 on the right-hand side of the equation.

34.4. f n+2 =/ n+1 +/„ where/ 0 = 1,/j = 1

The independent variable is n, the dependent variable is/. This is a second-order equation which is linear and
homogeneous. We note that there are two initial conditions. We also note that this relationship, coupled with the
initial conditions, generate a classical set of values known as the Fibonacci numbers (see Problems 34.7 and 3430).

34.5. y r = 9 cos y r_ 4
The independent variable is r, the dependent variable is y. This is a fourth-order equation. It is non-linear
because of the appearance of cos _ 4; it is a homogeneous equation because both terms contain the dependent
y r
variable.

34.6. 2" + Xn = x n+s
The independent variable is n, the dependent variable is x. This is an eighth-order linear difference equation.
It is non-homogeneous due to the 2" term.