Use Laplace Transforms to solve the following Initial Value Problems:

(i)

$\displaystyle \dot{y_1} = y_{2}$

$\displaystyle \dot{y_2} = -y_{1} + e^t$

with $\displaystyle y_{1}(0) = 0, y_{2}(0) = 0$

(ii)

$\displaystyle \dddot{y} = \begin{cases} t & \ t < 3 \\ 0 & \ t \geq 3 \end{cases}$ with $\displaystyle y(0) = 0$, $\displaystyle \dot{y}(0) = 0$ and $\displaystyle \ddot{y}(0) = 0$