difference equation

difference equation, mathematical equality involving the differences between successive values of a function of a discrete variable. A discrete variable is one that is defined or of interest only for values that differ by some finite amount, usually a constant and often 1; for example, the discrete variable x may have the values x₀ = a, x₁ = a + 1, x₂ = a + 2, . . ., x_n = a + n. The function y has the corresponding values y₀, y₁, y₂, . . ., y_n, from which the differences can be found:

Any equation that relates the values of Δy_i to each other or to x_i is a difference equation. In general, such an equation takes the form

Systematic methods have been developed for the solution of these equations and for those in which, for example, second-order differences are involved. A second-order difference is defined as