Consider the following initial value problem, in which an input of large amplitude and short duration has been idealized as a delta function. $$y'' + 4 \pi^2 y = 2 \pi \delta(t - 3), \quad \quad y(0) = 0, \quad y'(0) = 0.$$

1. Find the Laplace transform of the solution.
   
   $\displaystyle Y(s) = {\mathcal L}\left\lbrace y(t) \right\rbrace =$
   
   
2. Obtain the solution $y(t)$.
   
   $y(t) =$
   
   
3. Express the solution as a piecewise-defined function and think about what happens to the graph of the solution at $t = 3$.
   
   

   $y(t) =$ $\displaystyle \Bigg\lbrace$

   $\ \mbox{ if } \ 0 \leq t < 3,$ $\ \mbox{ if } \ 3 \leq t < \infty.$