$($a$)$ Consider the following function

Express $f(t)$ and $g(t)$ in terms of unit step functions, and then find its Laplace transforms.

$($b$)$ Use the Laplace transform to solve the given IVP

$($i$)y^{\text{'}\text{'}}+9y=4sint,y^{\text{'}}(0)=0$ and $y(0)=0.$

$($ii$)y^{\text{'}\text{'}}+y=2t{e}^t,y^{\text{'}}(0)=0$ and $y(0)=0.$