Diferenssiyhtälö

Diferenssiyhtälö, often translated as difference equation in English, is a mathematical equation that relates a sequence of numbers to its own subsequent values. In essence, it describes how a quantity changes from one discrete step to the next. Unlike differential equations which deal with continuous change, difference equations work with discrete time points or steps. The simplest form of a difference equation is an autonomous first-order equation, such as x_{n+1} = f(x_n), where x_n represents the value of the sequence at step n, and x_{n+1} is the value at the next step. The function f determines the rule for this transition.

Non-autonomous equations also exist, where the relation can depend on the step number itself, for instance, x_{n+1} = f(x_n, n). Higher-order difference equations involve multiple previous terms, such as x_{n+2} = f(x_{n+1}, x_n, n). These equations are fundamental in modeling discrete dynamical systems, population dynamics, economics, and various fields of science and engineering where phenomena are observed at regular intervals. Solving a difference equation involves finding a sequence that satisfies the equation for all relevant steps, often requiring initial conditions to determine a unique solution.