delta-de-dirac

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\title{Función Delta de Dirac}

\author{Juancarlos Cruz}

\date{\today}

\begin{document}

\maketitle

\subsection{Propiedades empleadas}
      
\begin{enumerate}
\item $\mathscr{L}\{\delta (t)\}=1$
\item $\mathscr{L}\{\delta (t - t_{0})\} = e^{-st}$
\end{enumerate}

\begin{enumerate}
\item $\mathscr{L}\{f(t)\}=F(s)$
\item $\mathscr{L}\{f'(t)\} = S\mathscr{L}\{f(t)\}-f(0)$
\item $\mathscr{L}\{f''(t)\} = S^2\mathscr{L}\{f(t)\}-Sf(0)-f'(0)$
\end{enumerate}


\section{Problemas propuestos}


\subsection{$y' - 3y = \delta(t-2) ; y_{(0)} = 0$}


\begin{enumerate}
\item Aplicamos Laplace a todos los miembros: \\
$s\mathscr{L}\{f(t)\} - f_{(0)} - 3\mathscr{L}\{f(t)\} = e^{-2s} $

\item Factorizamos $\mathscr{L}\{f(t)\}$ \\
$\mathscr{L}\{f(t)\}(s-3) = e^{-2s}$

\item Despejamos \\ 
$\mathscr{L}\{f(t)\} = \frac{e^{-2s}}{s-3}$ \\

\item Aplicamos Laplace inversa ($\mathscr{L}\{f(t)\}^{-1}$) \\ 
$f(t) = \mu(t-2)\mathscr{L}^{-1} (\frac{1}{s-3})_{t\to(t-2)}$

\item Resolvermos \\
$f(t) = \mu(t-2)e^{3(t-2)}$
\end{enumerate}


\subsection{$y' + y = \delta(t-1) ; y_{(0)} = 2$}
\begin{enumerate}
\item Aplicamos Laplace a todos los miembros: \\
$s\mathscr{L}\{f(t)\} - f_{(0)} + \mathscr{L}\{f(t)\} = e^{-s} $

\item Factorizamos $\mathscr{L}\{f(t)\}$ \\
$\mathscr{L}\{f(t)\}(s+1) - 2= e^{-s}$

\item Despejamos \\ 
$\mathscr{L}\{f(t)\} = \frac{e^{-s}+2}{s+1}$ \\

\item Aplicamos Laplace inversa ($\mathscr{L}\{f(t)\}^{-1}$) \\ 

$f(t) = \mathscr{L}^{-1} (\frac{e^{-s}}{s+1}) + 2\mathscr{L}^{-1} (\frac{1}{s+1}) $\\
$f(t) = \mu(t-1)\mathscr{L}^{-1} (\frac{1}{s+1})_{t\to(t-1)} + 2e^{-t}$

\item Resolvermos \\
$f(t) = \mu(t-1)e^{-(t-1)} + 2e^{-t}$
\end{enumerate}



\subsection{$y'' + y = \delta(t-2\pi) ; y_{(0)} = 0, y'_{(0)} = 1$}
\begin{enumerate}
\item Aplicamos Laplace a todos los miembros: \\
$s^2\mathscr{L}\{f(t)\} - sf_{(0)} - f'_{(0)} + \mathscr{L}\{f(t)\} = e^{-2\pi s } $

\item Factorizamos $\mathscr{L}\{f(t)\}$ \\
$\mathscr{L}\{f(t)\}(s^2+1) - 1= e^{-2\pi s}$

\item Despejamos \\ 
$\mathscr{L}\{f(t)\} = \frac{e^{-2\pi s}+1}{s^2+1}$ \\

\item Aplicamos Laplace inversa ($\mathscr{L}\{f(t)\}^{-1}$) \\ 

$f(t) = \mathscr{L}^{-1} (\frac{e^{-2\pi s}}{s^2+1}) + \mathscr{L}^{-1} (\frac{1}{s^2+1}) $\\
$f(t) = \mu(t-2\pi)\mathscr{L}^{-1} (\frac{1}{s^2+1})_{t\to(t-2\pi)} + sen(t)$

\item Resolvermos \\
$f(t) = \mu(t-2\pi)sen(t-2\pi) + sen(t)$
\end{enumerate}


\subsection{$y'' + 16y = \delta(t-2\pi) ; y_{(0)} = 0, y'_{(0)} = 1$}
\begin{enumerate}
\item Aplicamos Laplace a todos los miembros: \\
$s^2\mathscr{L}\{f(t)\} - sf_{(0)} - f'_{(0)} + 16\mathscr{L}\{f(t)\} = e^{-2\pi s } $

\item Factorizamos $\mathscr{L}\{f(t)\}$ \\
$\mathscr{L}\{f(t)\}(s^2+16)= e^{-2\pi s}$

\item Despejamos \\ 
$\mathscr{L}\{f(t)\} = \frac{e^{-2\pi s}}{s^2+16}$ \\

\item Aplicamos Laplace inversa ($\mathscr{L}\{f(t)\}^{-1}$) \\ 

$f(t) = \mu(t-2\pi)\mathscr{L}^{-1} (\frac{1}{s^2+16})_{t\to(t-2\pi)}$

\item Resolvermos \\
$f(t) = \mu(t-2\pi)sen(4(t-2\pi))$
\end{enumerate}

\end{document}