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.)

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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.

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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:

So .

Thus

Application of the conditions and gives (respectively)

==>

and

Which implies

<-- This correctly defines the solution.

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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?

-Dan