Problem #$6$: Consider the following initial value problem.

$y\text{'}+5y=\{\begin{array}{cc}0,& t1\\ 11,& 1t<mn>6</mn>\\ 0,& 6t<mi></mi>\\ y(0)=5\end{array}$

$($a$)$ Find the Laplace transform of the right hand side of the above differential equation.

$($b$)$ Let $y(t)$ denote the solution to the above differential equation, and let $Y(s)$ denote the Laplace transform of $y(t).$ Find $Y(s).$

$($c$)$ By taking the inverse Laplace transform of your answer to $($b$),$ the solution $y(t)$ can be written in the form $f(t)U(t-1)g(t)U(t-6)h(t).$ Enter the fungtion $f(t)$ into the answer box below.

$($d$)$ Referring to part $($c$)$ above, enter the function $g(t)$ into the answer box below.

$($e$)$ Referring to part $($c$)$ above, enter the function $h(t)$ into the answer box below.