1. The problem

Consider the difference equation $\displaystyle y[n] - \frac{5}{6}y[n-1] + \frac{1}{6}y[n-2] = \frac{1}{3}x[n-1]$.

What is the

(a)step responsefor the causal system satisfying this difference equation,

(b)general form of the homogeneous solutionof the difference equation?

(c)Consider a different system satisfying the difference equation that is neither causal nor LTI, but that has y[0] = y[1] = 1. Find theresponse of this system to x[n] = $\displaystyle \delta$[n].

2. Relevant equations and definitions

Oppenheim's definition of(b),the general form of the homogeneous solution:

$\displaystyle y_h[n] = \sum_{m=1}^{N} A_m z_m^{n} \text{ ................... (2.97)}$

3. The attempt at a solution

(a)step response, s[n]

I have calculated the impulse response $\displaystyle h[n] = 2( \frac{1}{2^n} - \frac{1}{3^n}) u[n]$, and put it to use in calculatings[n](where I get stuck):

$\displaystyle s[n] = \sum_{k=-\infty}^{\infty} x[k] h[n-k] = 2 \sum_{k=-\infty}^{\infty} u[n-k] ( \frac{1}{2^n} - \frac{1}{3^n}) u[n] = ?$

(b)general form of the homogeneous solution

I have no idea (similar problem as(a)in this question on MHF) ... the solution ought to be $\displaystyle y_h[n] = A_1 \frac{1}{2^n} + A_2 \frac{1}{3^n}$.

(c)response of a particular system to x[n] = $\displaystyle \delta$[n]

Confused as well. I do get particular values fory[n]:

y[-1] = 41/5,

y[0] = 1,

y[1] = 1,

y[2] = 2/3,

...

continuing in this manner (and hoping for a pattern to emerge) surely isn't the way to go?

The solution given is:$\displaystyle y[n] = 4 \frac{1}{2^n} - 3 \frac{1}{3^n} - 2 \frac{1}{2^n} u[-n-1] + 2 \frac{1}{3^n} u[-n-1]$. Other answers are possible.