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10 Problem B2.3 OGATA 4ed (Laplace Transform)

Let us calculate the Laplace transform of the following functions.





  1. \begin{displaymath}f(t)=sen(w\cdot t)\cdot cos(w\cdot t)\end{displaymath}


    Solucion:
    Let's change the function as follows:

    \begin{displaymath}f(t)=\frac{sen(2\cdot w\cdot t)}{2}\end{displaymath}


    Using the Laplace transform:

    \begin{displaymath}L(sen(w\cdot t))=\frac{w}{s^{2}+w^{2}}\end{displaymath}


    With what we get:

    \begin{displaymath}F(s)=\frac{w}{s^{2}+{(2\cdot w)}^{2}}\end{displaymath}





  2. \begin{displaymath}f(t)=t\cdot e^{-t}\cdot sen(5\cdot t)\end{displaymath}


    Solucion:
    We will use the following properties and transforms

    \begin{displaymath}L(t^{n}\cdot f(t))=(-1)^{n}\cdot \frac{d^{n}}{ds}(F(s))\end{displaymath}



    \begin{displaymath}L(e^{-a\cdot t}\cdot f(t))=F(s+a)\end{displaymath}



    \begin{displaymath}L(sen(w\cdot t))=\frac{w}{s^{2}+w^{2}}\end{displaymath}


    With what we get:

    \begin{displaymath}F(s)=(-1)\cdot \frac{d}{ds}\left( \frac{5}{(s+1)^{2}+5^{2}} \right)=\frac{10\cdot (s+1)}{((s+1)^2+5^{2})^{2}}\end{displaymath}


    Program with Scilab.

    t=0:0.1:5;
    ft=t.*exp(-t).*sin(5*t);
    s=%s;
    fs=10*(s+1)/((s+1)^2+5^2)^2;
    fs2=syslin('c',fs);
    fs1=csim('impulse',t,fs2);
    clf;
    subplot(2,1,1);
    plot2d(t,ft,2);
    xtitle('Phrasing');
    xgrid;
    subplot(2,1,2);
    plot2d(t,fs1,1);
    xtitle('Solution');
    xgrid;
    
    Image B2-4b