Chapter 7.6, Problem 31E

In Problems $25-32$, solve the given initial value problem using the method of Laplace transforms.

$y^{″}+5y^{\prime }+6y=g\left(t\right)$;

$y\left(0\right)=0$, $y^{\prime }\left(0\right)=2$,

where $g\left(t\right)=\{\begin{array}{l}0,0\le t<1\text{,}\\ t,1<t<5,\\ 1,5<t\end{array}$