ABSTRACT
The term difference equation sometimes refers to a specific type of recurrence relation. However, difference equation is frequently used to refer to any recurrence relation.
Let us now see how a difference equation is formulated. Consider the relation un = cn− 3, where c is an arbitrary constant. Now, un+1 = c(n+1)− 3. The required difference equation is obtained by eliminating c from un and un+1, which gives
un+1 = un + 3
n (n+ 1)− 3
⇒ nun+1 = (n+ 1)un + 3
Consider the linear difference equation of the form
c0un + c1un−1 + c2un−2 = f(n) (2.1)
The difference equation is homogeneous if f(n) = 0, otherwise it is nonhomogeneous. The order of the difference equation is the difference between the largest and smallest arguments appearing in the difference equation with unit interval. Thus, the order of equation (2.1) is 2.
