# Difference Equations

• Jul 10th 2007, 06:50 AM
topsquark
Difference Equations
I have been reading the recent difference equation post and a question occurred to me. (Please forgive me I know many techniques but little theory for linear algebra.)

I know how to solve something like
$a_{n + 2} - 3a_n = 2$

$na_{n + 2} - 3a_n = 0$
for example. Is there a general method to attack something like this?

(I don't know if this even has a "nice" solution, I'm just using it as a sample equation where one or more of the coefficients depends on n.)

-Dan
• Jul 11th 2007, 06:26 PM
mathisfun1
Particular cases are often easier than general ones. The example you listed in fact has a straightforward solution.

$a_{n+2} = \frac{3a_n}{n}$. If you work relation for small n, you find the general formula for a as $a_{2m} = \frac{3^m }{2^m m!}a_2, \ a_{2m-1} = \frac{3^m(2m)!}{2^mm!}a_1, \ m=1, 2, ...$ where $a_1$ and $a_2$ are arbitrary.
• Jul 11th 2007, 07:01 PM
ThePerfectHacker
Quote:

Originally Posted by topsquark
(I don't know if this even has a "nice" solution, I'm just using it as a sample equation where one or more of the coefficients depends on n.)

I am moving this to Discrete Math.

Think of Difference Equations as Differencial equations.

Does a general differencial equation have a "nice" solutions? I wish!

However, there is a method to finding them. A very powerful method is known as generating functions.