Imagine tracking the balance in your savings account. The amount doesn't change smoothly over time. Instead, it jumps up when you make a deposit or when interest is paid out at the end of the month. The change happens in distinct steps.

This is the world of difference equations. They describe relationships between values at different, separate points in time or space. Instead of looking at change continuously, like the speed of a car at any given instant, we look at how a quantity changes from one step to the next.

In these equations, we often use a subscript, like $y_n$, to represent the value of something at step 'n'. The 'n' could stand for the n-th day, the n-th year, or just the n-th iteration of a process. The equation then connects $y_n$ to its neighbors, like the previous value $y_{n-1}$ or the next value $y_{n+1}$.

For example, a simple model for population growth might state that the population next year ($P_{n+1}$) is a multiple of the population this year ($P_n$).

This equation tells us that the population grows by 5% each year. It doesn't tell us anything about the population halfway through the year; it only relates the population from one distinct year to the next.

The key distinction to grasp is between discrete and continuous change. Difference equations are the language of discrete systems. Their counterpart, differential equations, are the language of continuous systems.

A differential equation involves derivatives, like $dy/dt$, which represent an instantaneous rate of change. Think of the acceleration of a falling apple at a precise moment. A difference equation involves differences, like $y_{n+1} - y_n$, which represent change over a discrete interval.

Despite this core difference, the two are deeply connected. In fact, many methods for solving differential equations on a computer work by first approximating them with a difference equation. The computer then solves the problem step-by-step, which is a process it's very good at.

Like other types of equations, difference equations have a few key properties that help us classify and understand them.

Order: The order of a difference equation is the difference between the highest and lowest subscripts of the variable in the equation. For example, $y_{n+1} = 2y_n$ is a first-order equation because the difference between the subscripts ($n+1$ and $n$) is 1.

This equation is second-order:

The highest subscript is $n+2$ and the lowest is $n$. The difference is $(n+2) - n = 2$.

Linearity: A difference equation is linear if the unknown function and its values appear in a linear fashion. This means no terms like $y_n^2$, $\sqrt{y_n}$, or $y_n \cdot y_{n-1}$. The coefficients can depend on $n$, but not on $y_n$.

For instance, $y_{n+1} = n \cdot y_n + 4$ is linear. However, $y_{n+1} = y_n^2$ is nonlinear because of the squared term.

With these basics, you can start to recognize and describe the equations that govern step-by-step processes all around us.