Suppose we begin with the simplest possible differential equation, that relating current and voltage in an inductor:
We assume that one of the quantities, say is provided by some source. We need to discretize this continuous time equation. First sample at regular intervals (in the simplest case), to obtain a sequence , and assume that we will be obtaining a voltage sequence from it. There are many ways of approximating the differentiation operator;
This is probably the simplest way of discretizing a derivative, and is called a backwards difference.
yielding the scheme:
which in some sense behaves like the original continuous time equation.