content: \newpage
  \section{Exercises}
  
  \begin{exercise}
  \label{exc:LaplaceDk}
  Starting from the ``Laplace-D'' law
  \begin{spec}
  ℒ f' s  =  s * ℒ f s - f 0
  \end{spec}
  Derive a general formula for |ℒ {-"f^{(k)} "-} s|.
  \end{exercise}
  
  \begin{exercise}
    Find the Laplace transforms of the following functions:
  %
    \begin{enumerate}
      \item |\t ->  3*pow e (5*t)|
      \item |\t ->  pow e (alpha * t) - beta|
      \item |\t ->  pow e (t + frac pi 6)|
    \end{enumerate}
  
  \end{exercise}
  
  \begin{exercise}  \leavevmode
    \begin{enumerate}
     %
      \item Show that: \label{exc:sincosexp}
        \begin{itemize}
          \item |sin t = frac 1 (2*i) * (  pow e (i*t) -  pow e (-i*t)  )|
          \item |cos t = frac 1 2     * (  pow e (i*t) +  pow e (-i*t)  )|
        \end{itemize}
  
      \item Find the Laplace transforms |ℒ sin| and |ℒ cos| using the
       transform for the exponentials and the result from \cref{exc:sincosexp}.
  
    \end{enumerate}
  \end{exercise}
  
  % TODO (by DaHe): Add some exercises on inverse Laplace transforms and partial
  % fraction decomposition
  
  %if lectureNotes
  \subsection{Exercises from old exams}
  %endif
  
  \begin{exercise}
    \textit{From exam 2016-03-15}
  
    Consider the following differential equation:
  
    $$f''\, t - 2 ⋅ f'\, t + f\, t = e^{2 ⋅ t},\quad f\, 0 = 2,\quad f'\, 0 = 3$$
  
    Solve the equation using the Laplace transform.  You
    should need only one formula (and linearity):
  
    $$\mathcal{L}\, (\lambda t.\, e^{\alpha⋅t})\, s  = 1 / (s - \alpha)$$
  \end{exercise}
  
  
  \begin{exercise}
    \textit{From exam 2016-08-23}
  
    Consider the following differential equation:
  
    $$f''\, t - 5 ⋅ f'\, t + 6 ⋅ f\, t = e^t,\quad f\, 0 = 1,\quad f'\, 0 = 4$$
  
    Solve the equation using the Laplace transform.  You
    should need only one formula (and linearity):
  
    $$\mathcal{L}\, (\lambda t.\, e^{\alpha⋅t})\, s  = 1 / (s - \alpha)$$
  \end{exercise}
  
  \begin{exercise}
    \textit{From exam 2016-Practice}
  
    Consider the following differential equation:
  
    $$f''\, t - 2 ⋅ f'\, t + f\, t - 2 = 3 ⋅ e^{2 ⋅ t},\quad f\, 0 = 5,\quad f'\, 0 = 6$$
  
    Solve the equation using the Laplace transform.  You
    should need only one formula (and linearity):
  
    $$\mathcal{L}\, (\lambda t.\, e^{\alpha⋅t})\, s  = 1 / (s - \alpha)$$
  \end{exercise}
  
  \begin{exercise}
    \textit{From exam 2017-03-14}
  
    Consider the following differential equation:
  
    $$f'' \,t + 4⋅f\, t = 6⋅\cos\, t, \quad f\, 0 = 0,\quad f'\, 0 = 0$$
  
    Solve the equation using the Laplace transform.  You
    should need only two formulas (and linearity):
  
    $$\mathcal{L}\, (\lambda t.\, e^{\alpha⋅t})\, s  = 1 / (s - \alpha)$$
  
    $$2 ⋅ \cos \,t = e^{i⋅t} + e^{-i⋅t}$$
  \end{exercise}
  
  
  \begin{exercise}
    \textit{From exam 2017-08-22}
  
    Consider the following differential equation:
  
    $$f'' \,t -3 \sqrt{2} ⋅ f' \, t + 4 ⋅ f \, t = 0, \quad f\, 0 = 2,\quad f'\, 0 = 3 \sqrt{2}$$
  
    Solve the equation using the Laplace transform.  You
    should need only one formula (and linearity):
  
    $$\mathcal{L}\, (\lambda t.\, e^{\alpha⋅t})\, s  = 1 / (s - \alpha)$$
  \end{exercise}
  