Consider the difference equation (P2.32-1)...

Consider the difference equation

(P2.32-1)

and suppose that

(P2.32-2)

Assume that the solution y[n] consists of the sum of a particular solution y_p [n] to eq. (P2.32-1) and a homogeneous solution y_h [n] satisfying the equation

(a) Verify that the homogeneous solution is given by

(b) Let us consider obtaining a particular solution y_p [n] such that

By assuming that y_p [n] is of the form and substituting the above difference equation, determine the value of B.

(c) Suppose that the LTI system described by eq. (P2.32-1) and initially at rest has as its input the signal specified by eq. (P2.32-2). Since x[n] = 0 for n < 0, we have that y[n] = 0 for n < 0. Also, from parts (a) and (b) we have that y[n] has the form

for n ≥ 0. In order to solve for the unknown constant A, we must specify a value for y[n] for some n ≥ 0. Use the condition of initial rest and eqs. (P2.32-1) and (P2.32-2) to determine y[0]. From this value determine the constant A. The result of this calculation yields the solution to the difference equation (P2.32-2) under the condition of initial rest, when the input is given by eq. (P2.32-2).