Consider the initial value problem $$y'' + 4 y = g(t), \quad \quad y(0) = 0, \quad y'(0) = 0,$$ where $\ g(t) = \left\lbrace \begin{array}{ l l } t & \mbox{ if } 0\leq t < 4 \\ 0 & \mbox{ if } 4 \leq t < \infty. \end{array} \right.$

1. Take the Laplace transform of both sides of the given differential equation to create the corresponding algebraic equation. Denote the Laplace transform of $y(t)$ by $Y(s)$. Do not move any terms from one side of the equation to the other (until you get to part (b) below).
   
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2. Solve your equation for $Y(s)$.
   
   $\displaystyle Y(s) = {\mathcal L}\left\lbrace y(t) \right\rbrace =$
   
   
3. Take the inverse Laplace transform of both sides of the previous equation to solve for $y(t)$.
   If necessary, use $h(t)$ to denote the Heaviside function $\displaystyle h(t) = \begin{cases} 0 & \text{if } t < 0 \\ 1 & \text{if } 0 \leq t \end{cases}$.
   
   $y(t) =$