Methods for Inhomogeneous Difference Equations
I believe that the term "inhomogeneous difference equation" describes the following problem:
with the conditions and . (This problem has no relevance to anything, I just made it up.)
TPH gave a method of solution for this.
The homogeneous version of this difference equation is
and has the characteristic equation
So m = 1 is a double root. Thus
as the homogeneous solution.
We need a particular solution. So let . Then , and substitution into the original equation gives:
which is impossible.
Soroban also gave a method. Consider the related difference equation:
When we subtract the original equation
from this we get the homogeneous difference equation:
The characteristic equation here is
and has m = 1 as a triple root. So the solution of the homogeneous equation is
Now we substitute this solution into the original equation:
After a number of cancellations:
Application of the conditions and gives (respectively)
<-- This correctly defines the solution.
My question (long in coming) is why don't both of these methods work? Is there an assumption in TPH's method that makes it unviable for this specific problem? And, if so, is Soroban's the most general method?