Exercise $1\mathrm{.}2\mathrm{.}3$ Suppose that $x(t)$ is the input to a linear time$-$invariant system with corresponding output $y(t).$ Prove that the system has the following properties:

a$.$ An input $x_1(t)={\displaystyle \underset{0}{\overset{t}{}}}x()d$ to the same system yields an output of $y_1(t)={\displaystyle \underset{0}{\overset{t}{}}}y()d$

b$.$ An input $x_1(t)=\frac{dx(t)}{dt}$ to the same system yields an output of $y_1(t)=\frac{dy(t)}{dt}$

Exercise $1\mathrm{.}2\mathrm{.}4$ Compute the unit$-$pulse response $h(n)$ for all integers for each of the following discrete$-$time systems:

a$.y(n+1)+y(n)=x(n)$

b$.y(n+1)+2y(n)=x(n+1)-2x(n)$