Determine if $ y[n] = ny[n-1] + x[n]$ is linear time invariant and BIBO stable Check if the following system is linear time invariant and BIBO stable..
$$
y[n] = ny[n-1] + x[n]
$$

for $n\ge 0$. We are also given that the system is at rest (i.e. $y[−1] = 0$).

I know that to show linearity, we show that 
$
H[ax_1 + bx_2] = aH[x_1] + bH[x_2]
$
except that the question has a $y[n-1]$ on the RHS which is throwing me off.

Edit: My solution:
$
LHS = $ $$H[ax_1[n] + bx_2[n]] 
$$$$
= n(ay_1[n−1] + by_2[n−1]) + (ax_1[n]+bx_2[n])
$$$
 RHS =$$$  aH[x_1] + bH[x_2] 
$$$$
=  any_1[n−1]+ax_1[n]+bny_2[n−1]+bx_2[n]
$$
So $
 RHS = LHS
$

Edit 2: New issue when I try and show time invariance.

If $ x_2[n] = x1[n-k] $ then prove $y_2[n]  = y_1[n-k]$.

BUT $RHS= $
$$
(n-k)y_1[n-k-1] + x_1[n-k]
$$ And $LHS = $
$$ny_2[n-1] + x_2[n] = ny_2[n-1] + x_1[n-k]$$
I'll give you some hints that hopefully will allow you to do your homework yourself.

In order to check linearity, write down $3$ difference equations: one for the response $y_1[n]$ to an input signal $x_1[n]$, one for the response $y_2[n]$ to an input signal $x_2[n]$, and one for the response $y_3[n]$ to an input signal $ax_1[n]+bx_2[n]$. Multiply the first difference equation by $a$, the second one by $b$, and add them. If the resulting difference equation for the sequence $ay_1[n]+by_2[n]$ is the same as for $y_3[n]$, then you can conclude $y_3[n]=ay_1[n]+by_2[n]$, and, consequently, the system must be linear.

For checking time-invariance, write down the difference equation for the response $y_2[n]$ to the input $x[n-k]$. Then replace in the original difference equation the index $n$ by $n-k$ and check if the two difference equations are the same. If they are, the system is time-invariant.

To answer the question about BIBO stability, ask yourself what the output signal will be for $x[n]=\delta[n]$ and draw your conclusion.



Now that you've figured out the solution I would like to point out that you can also find an explicit (non-recursive) expression for $y[n]$:

$$\begin{align} y[0]&amp;=x[0]\\
y[1]&amp;=1\cdot x[0]+x[1]\\
y[2]&amp;=2\cdot 1\cdot x[0]+2\cdot x[1]+x[2]\\
y[3]&amp;=3\cdot 2\cdot 1\cdot x[0]+3\cdot 2\cdot x[1]+3\cdot x[2]+x[3]\\
&amp;\vdots\\
y[n]&amp;=\sum_{k=0}^n\frac{n!}{k!}x[k],\qquad n\ge 0\tag{1}
\end{align}$$

from which it is very straightforward to show that the system is linear and time-varying. Furthermore, from $(1)$ it should be clear that the system is not BIBO-stable.