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Math Help - HELP LAPLACE TRANSFORM!!!!

  1. #1
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    HELP LAPLACE TRANSFORM!!!!

    Hey. My exam is in a few hours so it'd be cool if someone could show me this. I can guarantee you it won't hinder my understanding by just "giving me the answer"... nothing would benefit me more at this point haha.

    Anyway. I don't even know if it's possible but can you find a Laplace transform of this in terms of Y(s):

     \mathcal L (y(t).sin(\omega t))

    It's part of a bigger question where I have to find the transfer function of the differential equation:

     \ddot{x} + \zeta \omega_n \dot{x} + \omega_n^2 x = y(t)sin(\omega t)

    Assuming zero initial conditions: Hence my LHS is:

     (s^2 + \zeta \omega_n s + \omega_n^2)X(s) = \mathcal L (y(t).sin(\omega t) )

    The transfer function is given by the ratio of the frequency-domain input to the frequency-domain output. The input being Y(s), and the output being X(s).

    In short I need to get Y(s) on the RHS of that equation using the laplace transform integral I'd assume, and then manipulate the equation to get \frac{X(s)}{Y(s)}

    (NOTE:  \omega_n, \omega,  \zeta are all constants.[/tex]

    Many thanks.
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  2. #2
    Super Member
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    Aug 2008
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    I saw that yesterday but I could only get it in terms of Y(s-i\omega) since:

    L\left\{y(t)\sin(\omega t)\right\}=\text{Im}\left( L\left\{e^{i\omega t} y(t)\right\}\right)=\text{Im}\left\{Y(s-i\omega)\right\}
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