Consider the initial value problem $$y'' + 4 y = 12 t, \quad \quad y(0) = 3, \quad y'(0) = 8.$$

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).
   
   $=$ help (formulas)
   
   
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 $y(t)$.
   
   $y(t) =$