Homogeneous Recurrence Relation:
A homogeneous recurrence relation is one in which each term is defined as a linear combination of previous terms with no additional (non-zero) function or sequence added.
Non-Homogeneous Recurrence Relation:
A non-homogeneous recurrence relation is one in which each term is defined as a linear combination of previous terms plus an additional function (often called the non-homogeneous part).
