There are two different types of Difference equations: a) Homogeneous Difference Equations and b) Non-homogeneous Difference Equations. This step-by-step guide will help you how to solve these problems that will MOST DEFINITELY come up in one of Dr. Morley's Quizzes in Math 2602.

Remember
This is about Difference Equations and not Differential Equations... differential equations is a whole separate course altogether.

The homogeneous difference equations topic is covered in 5.5 while the non-homogeneous difference equations is covered in 5.7. There are couple of examples that are useful and the text also explains what difference equations are.

1. Find the general solution for the homogeneous difference equation:

a_n = -3a_n-1 - 2a_n-2

2. Step 1: Substitute rⁿ for a_n

rⁿ = -3a^n-1 - 2a^n-2

3. Step 2: Set Equal to Zero

rⁿ + 3a^n-1 + 2a^n-2 = 0

4. Step 3: Substitute as follows (rⁿ=r², r^n-1=r, ...)

r² + 3r + 2 = 0
(This is called the Characteristic Equation)

5. Step 4: Solve for r

(r + 2)(r + 1) = 0
r = -2, -1

6. Step 5: There are two geometric solutions to this example:

Solution 1: a_n = (-2)ⁿ
Solution 2: a_n = (-1)ⁿ

7. Step 6: The general solution is the linear combination of the two solutions:

a_n = A(-2)ⁿ + B(-1)ⁿ

8. Tips: If there are initial conditions, you simply need take general solutions and make them equal to the initial conditions. Then, solve the systems of equations to find the coefficients A and B. This will give you a specific solution for the initial conditions given. (See page 441).

Non-homogeneous difference equations are tougher to cover because there are varying types of problems. The step-by-step example below is of medium difficulty.

1. Find the general solution for the homogeneous difference equation:

a_n = 3a_n-1 + n2ⁿ

2. Step 1: Set Equal to the non-homogeneous part of the difference equation

a_n - 3a_n-1 = n2ⁿ

3. Step 2: Substitute (A + B)2ⁿ for a_n

(A + B)2ⁿ - 3(A + B(n - 1))2^{n - 1} = n2ⁿ

4. Step 3: Divide by 2^{n - 1}

2A + 2Bn - 3A - 3Bn + 3B = 2n
-A + 3B = 0
-Bn = 2n

5. Step 4: Solve for A, and B

A = -6, B = -2

6. Step 5: Plug A and B back into difference equation for the general solution

a_n = (-6 - 2n)2^{n + 1} + A3ⁿ

7. Tips: The most confusing part of non-homogeneous difference equations is Step 3 and all the algebra you have to do to find the general solution. Be careful of these two things and you should be able to figure out the general solutions to these difference equations without a problem.

This really is a tip for everyone at Tech and not just people in Math2602. Morley has iterated the importance of this numerous times in lecture, coweb, quizzes, and finals... DON'T PANIC. It doesn't matter what you did or didn't do before a quiz or an exam; when you panic, you will have a difficult time concentrating on the problem at hand. Unfortunately, I've had multiple personal experiences in this matter and I can tell you, the results aren't pretty. Whenever you feel panicky about an exam, take a deep breath and work slow. Also, make sure you stop studying 30-60 minutes prior to a test just to clear your mind. Cramming in the last minute will only help you panic...