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However, on the pages you mentioned they appear to make a difference between the two. And the difference is as follows. The zero-input response is the response which is caused by non-zero initial conditions. It only depends on the system properties and on the values of the initial conditions. The zero-input response becomes zero if the initial conditions are zero.

However, on the pages you mentioned they appear to make a difference between the two. And the difference is as follows. The zero-input response is the response which is caused by non-zero initial conditions. It only depends on the system properties and on the values of the initial conditions. The zero-input response is zero if the initial conditions are zero.

Also note that it is the shape of the zero-state response that depends on the poles of the system as well as on the poles of the input signal transform. All other responses mentioned here only depend on one of the two sets of poles. The shapes of of the zero-input response and of the natural response depend only on the system's poles, whereas the shape of the forced response is determined by the poles of the input signal. The expression for $y[n]$ quoted in your question from Proakis and Manolakis is the zero-state response (because the system is initially relaxed), and the first sum is the forced response, and the second sum is the natural response. Since the zero-input response is zero in this case, the sum of natural response and forced response (i.e., the total response) equals the zero-state response

Also note that it is the shape of the zero-state response that depends on the poles of the system as well as on the poles of the input signal transform. All other responses mentioned here only depend on one of the two sets of poles. The shapes of the zero-input response and of the natural response depend only on the system's poles, whereas the shape of the forced response is determined by the poles of the input signal. The expression for $y[n]$ quoted in your question from Proakis and Manolakis is the zero-state response (because the system is initially at rest), and the first sum is the forced response, and the second sum is the natural response. Since the zero-input response is zero in this case, the sum of natural response and forced response (i.e., the total response) equals the zero-state response

Again using the above example will hopefully clarify this. For an exponential forcing signal, the standard (and most straight-forward) way to obtain a particular solution is choosing a scaled version of the forcing function:

$$y_p[n]=Ab^n\tag{A1}$$

(for the sake of simplicity I leave out the unit step $u[n]$, assuming that we consider $n\ge 0$, unless we talk about the initial condition). The constant $A$ is determined by plugging $(A1)$ into the difference equation:

$$Ab^n+aAb^{n-1}=b^n$$

giving $A=\frac{b}{a+b}$. The general form of the homogeneous solution is

$$y_h[n]=B(-a)^n\tag{A2}$$

Of course $y_h[n]=0$ (i.e., $B=0$) is one specific solution, but that's not the one we're looking for. We need to determine the constant $B$ in such a way that the sum of the particular and the homogeneous solution satisfies the initial condition:

$$y[-1]=y_p[-1]+y_h[-1]=\frac{A}{b}-\frac{B}{a}$$

From this equation we get

$$B=\frac{a}{a+b}-ay[-1]$$

which shows that the homogeneous solution we need is non-zero if $y[-1]=0$. $y_p[n]$ and $y_h[n]$ found in this way are identical to the forced response and the natural response, respectively, as shown in $(4)$ and - implicitly - in $(2)$.