Particular cases are often easier than general ones. The example you listed in fact has a straightforward solution.
. If you work relation for small n, you find the general formula for a as where and are arbitrary.
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
But what about
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.)
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.
You can learn more about them here.