Difference equation: (1) step response, (2) homogeneous solution, (3) ...

**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 response** for the causal system satisfying this difference equation,

**(b)** **general form of the homogeneous solution** of 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 the **response 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 calculating **s[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 for **y[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: Quote:

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

Re: Difference equation: (1) step response, (2) homogeneous solution, (3) ...

Bumping in hope of an answer.

Re: Difference equation: (1) step response, (2) homogeneous solution, (3) ...

Found the method for solving **(b)**; still need help with **(a)** and **(c)**.

Re: Difference equation: (1) step response, (2) homogeneous solution, (3) ...

Got **(a)**, still plead for help with **(c)**.