Solution:

An example of the class of linear constant-coefficient difference equations is the accumulator system defined by,

$$ y[n]=\sum_{k=-\infty}^n x[k]....(1) . $$

To show that the input and output satisfy a differential equation, note that we can write the output for $n-1$ as

$$ y[n-1]=\sum_{k=-\infty}^{n-1} x[k]....(2) $$

By separating the term $x[n]$ from the sum, we can rewrite it as

$$ y[n]=x[n]+\sum_{k=-\infty}^{n-1} x[k].....(3) $$

Substituting Eq. (1) into Eq. (2) yields,

$$ y[n]=x[n]+y[n-1], $$

from which the desired form of the difference equation can be obtained by grouping all the input and output terms on separate sides of the equation:

$$ y[n]-y[n-1]=x[n] . $$

Thus, we have shown that, in addition to satisfying the defining relationship of Eq. (2), the input and output satisfy a linear constant-coefficient difference equation of the form Eq. (3), with $N=1, a_0=1, a_1=-1, M=0$, and $b_0=1$.

The difference equation in Eq. (3) gives us a better understanding of how we could implement the accumulator system.

According to Eq. (4), for each value of n, we add the current input value x[n] to the previously accumulated sum $y[n-1]$. This interpretation of the accumulator is represented in block diagram form in Figure.