A difference equation is an equation the solution of which is a sequence. So it might be a bit more suggestive to write your equation like this:

Instead of the differential operator you have the "shift operator" (that shifts the sequence by a certain fixed number of places). So your difference equation is a homoegenous second-order linear difference equation.

Because it is linear and homogeneous the general solution is a linear combination of certain base solutions. The base solutions can be found by setting

, and pluging this into your equation:

, the non-trivial solutions of which are

.

Thus the general solution of your homogeneous linear difference equation is

.

Now bring in the "initial conditions" to figure out what

should be...