$($a$)$ Derive the laplace transform of unit impulse function.

$($b$)$ Discuss the initial value theorem and find the initial value $x(0)$ of the function that has the following transform :

$x(5)=\frac{s^3-6s^2+9s-8}{s(s-2)(s^3+2s^2-s-2)}$

$($c$)$ Solve the following equation $\frac{dx}{dt}={\displaystyle \underset{0}{\overset{1}{}}}x(t)dt-t$ where $x(0)=3.$