Help Solving a Difference Equation I have an exam in my signals and systems class in a couple of days, and I'm unsure how to go about solving this practice problem. The question is as follows:

Consider a discrete time system whose input and output are related by the following difference equation.

$$  y[n] =
\begin{cases}
(1/4)y[n-1] + x[n],  &amp; \text{if $n&gt;=0$} \\
(1/4)y[n-1] + x[n] + x[n+1]*\cos(n*u[n] + \pi/2), &amp; \text{otherwise}
\end{cases}
$$

Use recursion to find an expression for the zero-state output, y[n], when the input is given by
$$x[n] = δ[n + m]$$
where m is an integer

I'm having a bit of trouble with the fact that the equation extends to before 0; usually to solve one of these by recursion we'd have something like y[-1] = 0, which would allow us to go ahead and compute more ys ... but in this case that seems to be impossible.

The only thing I managed to notice was that cos expression is irrelevant because of the unit step function; it's all 0 before 0, and that whole term goes away.Also recursively the y[n-1] term seems to divide by infinity ... but I don't know where to take it from there.
You seem to have figured out that the system is actually given by

$$y[n]=\frac14 y[n-1]+x[n],\quad \forall n\tag{1}$$

Since you're looking for the zero-state output, you can simply compute the system's impulse response $h[n]$ by applying $x[n]=\delta[n]$ as an input signal (with $y[n-1]=0$), and compute the response to any other input signal using this impulse response. For $x[n]=\delta[n+m]$, the response is simply $h[n+m]$.

The recursion for the impulse response is

$$\begin{align}h[0]&amp;=1\\
h[1]&amp;=\frac14\\
h[2]&amp;=\left(\frac14\right)^2\\
&amp;\vdots\end{align}\\$$

I'm sure you can take it from here and derive a general expression for $h[n]$.