In Section 2.5, we stated that the solution to the homogeneous difference equation...

In Section 2.5, we stated that the solution to the homogeneous difference equation

is of the form

with the A_m’s arbitrary and the z_m’s the N roots of the polynomial

(a) Determine the general form of the homogeneous solution to the difference equation

(b) Determine the coefficients A_m in the homogeneous solution if y[−1] = 1 and y[0] = 0.

(c) Now consider the difference equation

If the homogeneous solution contains only terms of the form of Eq. (P2.50-1), show that the initial conditions y[−1] = 1 and y[0] = 0 cannot be satisfied.

(d) If Eq. (P2.50-2) has two roots that are identical, then, in place of Eq. (P2.50-1), y_h[n] will take the form

where we have assumed that the double root is z₁. Using Eq. (P2.50-4), determine the general form of y_h[n] for Eq. (P2.50-3). Verify explicitly that your answer satisfies Eq. (P2.50-3) with x[n] = 0.

(e) Determine the coefficientsA₁ andB₁ in the homogeneous solution obtained in part (d) if y[−1] = 1 and y[0] = 0.