Question $5$: $(2$ marks$)$

A discrete system is represented by the following difference equation:

$y[n+3]-2y[n+1]+3y[n]=x[n+2]-x[n+1]$

with the initials conditions $y[-3]=-3,y[-2]=-2,$ and $y[-1]=-1.$ Solve this difference equation recursively for the input $x[n]=r[n]$ by finding the system output $y[n]$ at $n=0,1,2,3.$

Question $6$: $(2$ marks$)$

Determine the output ${?}^{?}$

$u[n]$ of the discrete$-$time system whose input and impulse response are given by $x[n]=(4{)}^{u^{}}u[n-2]$ and $h[n]=(2{)}^{\text{'}}u[n],$ respectively. Use direct expression.

Hint: ${\displaystyle \underset{k=m}{\overset{n}{}}}{r}^k=\frac{r^{n+1}-r^m}{r-1},r1.$