$1)$ Find solution of the differential equation below for $y(0)=2.$

$\frac{dy}{dx}+\frac{y}{2}=\frac{3}{2}$

$2)$ Find solution of the differential equation; $t^{\text{'}\text{'}}+5t^{\text{'}}+6t=0$ for the initial conditions $t(0)=2$ and $t^{\text{'}}(0)=3.$

$3)$ Assuming A and a are constants, determine the Laplace transform of the function $f(t)=A{e}^{at}$ using the definition of the transform.

$4)$ Find solution of the differential equation; $x^{\text{'}\text{'}}-5x^{\text{'}}+6x=6t-4$ using only Laplace transforms operations for the initial conditions $x(0)=1$ and $x^{\text{'}}(0)=2.$