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\begin{document}

\begin{center}
  \Large{13. Gyak.}  

  Laplace transzformált alkalmazása
  \end{center}
  
  \Large
  
\medskip
\begin{enumerate}
  \item $\mathcal{L}(2t \cdot e^{3t})(s)=?, \ \mathcal{L}(f(t))(s)=e^{-2s} \frac{3}{s(s+1)}, \ f(t)=?$
  \item $\mathcal{L}(t^2\sinh 2t)(s)=?, \ \mathcal{L}(f(t))(s)=\frac{e^{-2s}-e^{-4s}}{s}, \ f(t)=?$
  \item $\mathcal{L}\left( \int_0^t \tau \cos \tau \operatorname{d}\tau\right)(s)=?$
  \item $\mathcal{L}\left( \frac{1}{t} \sin 2t \right)(s)=?$
  \item $2\ast e^{-t}=?, \ \mathcal{L}\left( 2\ast e^{-t} \right)(s)=?$
  \item $\sin t\ast t=?, \ \mathcal{L}\left( \sin t\ast t \right)(s)=?$
  
  \item $y'(t)+\int_0^t y(\tau)\cosh(t-\tau) \operatorname{d}\tau=0, \ y(0)=1, \ y(t)=?$
  \item $\sinh t-t=\int_0^t \cosh \tau \cdot x(t-\tau)\operatorname{d}\tau=0, \ x(t)=?$
  \item $\ddot{x}+\dot{x}-12 x=f(t), \ x(0)=\dot{x}(0)=0$
   \[
    f(t)=\begin{cases}
	       1-t & \text{ ha } 0<t<2 \\
		   0 & \text{különben}
	     \end{cases}
   \]
\end{enumerate}

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