Difference Equation¶

Definition¶

A difference equation of order $k\ge1$ for a sequence $(x_t)_{t\ge0}$ is a relation

x_{t+k}=f\big(t,\,x_t,\,x_{t+1},\,\ldots,\,x_{t+k-1}\big),

for $t=0,1,2,\ldots$ $0, 1, 2, \dots$ , in which $x_t$ actually occurs; the order is the gap between the largest and the smallest time index. A solution is a sequence satisfying the relation for every $t\ge0$ $0$ .

The equation is linear if it has the form

x_{t+k}+a_{k-1}(t)\,x_{t+k-1}+\cdots +a_1(t)\,x_{t+1}+a_0(t)\,x_t=g_t,

with $a_0(t)\ne0$ for all $t$ . It is homogeneous if $g_t=0$ for all $t$ , and has constant coefficients if no $a_i$ depends on $t$ . A constant solution $x_t=x^*$ is an equilibrium (steady state).

Interpretation¶

A difference equation is a rule that produces the next value of a quantity from the current and a few past values, as in a savings balance, a price, or a population observed once per period. Time is discrete; the continuous-time counterpart is a differential equation, studied from Week 3.

The Fibonacci sequence 0, 1, 1, 2, 3, 5, 8 drawn as a row of boxes, the two initial values shaded, with arrows showing each new term computed from the two terms before it.

The figure runs the Fibonacci example $x_{t+2}=x_{t+1}+x_t$ $x_{t}$ . The two shaded initial values $x_0=0$ $0$ , $x_1=1$ are given; every later box is filled by the rule from the two boxes before it, so $x_2=x_1+x_0=1$ and, further on, $x_5=x_4+x_3=5$ $5$ . Because the equation has order $2$ , exactly two starting values are needed, and changing them produces a different solution.

The name comes from the forward difference $\Delta x_t=x_{t+1}-x_t$ $x_{t}$ : the equation $\Delta x_t=0.1x_t$ is the same as $x_{t+1}=1.1x_t$ $1.1 x_{t}$ . Any relation between $x_t$ and its differences can be rewritten with shifted terms.

The order counts how many consecutive past values the rule needs, not the largest index. In $x_{t+3}=x_{t+1}$ the smallest index is $t+1$ $1$ , so with $s=t+1$ it reads $x_{s+2}=x_s$ $x_{s}$ , an equation of order $2$ . For a homogeneous linear equation the solution set is a vector space (for an inhomogeneous one it is a shifted copy of that space, not a vector space), which is why linear algebra applies.

Key Results¶

For an equation in the explicit form above, any choice of the $k$ initial values $x_0,\ldots,x_{k-1}$ determines exactly one solution, computed recursively.
The solutions of a homogeneous linear equation of order $k$ form a $k$ -dimensional subspace of the space of real sequences; see basis of solutions.
For a linear equation, every solution equals one particular solution plus a solution of the homogeneous equation.
An equilibrium of $x_{t+1}=f(x_t)$ is a fixed point, $x^*=f(x^*)$ $f (x^{*})$ .
A linear equation of order $k$ can be rewritten as a first-order system in $\mathbb{R}^k$ .

Examples¶

First order. $x_{t+1}=2x_t$ with $x_0=3$ gives $x_t=3\cdot2^t$ $2^{t}$ : $3,6,12,24,\ldots$

Fibonacci. $x_{t+2}=x_{t+1}+x_t$ is linear, homogeneous, of order $2$ with constant coefficients. With $x_0=0$ $0$ , $x_1=1$ the recursion gives $0,1,1,2,3,5,8,\ldots$ ; other initial values give other solutions.

Nonlinear. The logistic map $x_{t+1}=r\,x_t(1-x_t)$ has order $1$ but is not linear. Its equilibria solve $x^*=r x^*(1-x^*)$ $x^{*})$ , so $x^*=0$ or $x^*=1-1/r$ (for $r\ne0$ $0$ ).

Prerequisites¶

Sources¶

Math V: Advanced Linear Algebra · Week 2

Aliases: recurrence relation, recurrence equation, discrete-time dynamical system, discrete dynamical system Keywords: x_{t+1}, order of a difference equation, linear difference equation, homogeneous difference equation, initial values, Fibonacci recurrence Symbols: delta, real numbers

Last updated: 22 October 2026 19:04