Step 1: What you want to do is drop the constant, the same as simply making
the constant zero to get the homogeneous difference equation. Then you get
a general solution to that homogenous difference equation.
Step 2: Then go back and come up with a trial solution to the non-homogenous
difference equation, the style of the trial solution being based on the kind
of terms that are creating the non-homogeneous part (this will probably be
the hardest part, so read that section of material carefully). Then put in
particular values that you know solve the difference equation, specific
values you got from working with the difference equation such as what xo
is, what x1 is, allowing you to determine what the constants in your trial
solution need to be. This gives you your particular solution.
Step 3: As the last step you combine your general solution and your
particular solution to get your complete solution for this specific
problems, and of course verify that when applied it creates the correct
values for x0, x1, .... xn.
Overall you will have started with a "difference equation" expressing how to
get x n+1 if you know x n, and you are ending with a formula that will
calculate any x n by just knowing n. The formula allows you to calculate
any x n term without working out all of the terms ahead of it.