An equation whose solution is a sequence (e.g. 0, 1, 1, 2, 3, 5, 8, .... See Fibonacci numbers & Bernoulli's method). Since a sequence is merely a function defined on a set of integers the theory of difference equations is often similar to the theory of differential equations.
They play an important role in numerical analysis and also in combinatorial analysis, economics, ecology and probability.
The difference equation for the sequence above (where s0 = 0, s1 = 1, etc..) is:
sn - sn-1 - sn-2 = 0. (See Bernoulli's method.)
The difference equation sn - sn-1 - n = 0 (positive integers) has a solution sn = n(n + 1)/2 (there are other solutions).
Order of the equation
sn + sn-1 = 0, for example, is called an equation of order 1.
sn + sn-1 + sn-2 = 0 is of
order 2, etc..
- sn - sn-1 - n2sn-2 = 1 is called "linear."
- sn - 2(sn-1)2 = 0 is not.