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Math Help - My curiosity needs satisfying

  1. #1
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    My curiosity needs satisfying

    I had a maths exam last Monday and the only question I couldn't do is below. Can anyone shed any light?

    Laplace transform the following:

    1/(s^2+s+5/4)
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  2. #2
    Eater of Worlds
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    Don't you mean the inverse LaPlace?.

    In that event, \frac{sin(t)}{\sqrt{e^{t}}}
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  3. #3
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by boab View Post
    I had a maths exam last Monday and the only question I couldn't do is below. Can anyone shed any light?

    Laplace transform the following:

    1/(s^2+s+5/4)

    Inverse Laplace:

    \mathcal{L} ^{-1} \left\{\frac{1}{s^2+s+\frac{5}{4}}\right\}
    \implies \mathcal{L}^{-1} \left\{\frac{1}{\left(s^2+s+\frac{1}{4}\right)+\fr  ac{5}{4}-\frac{1}{4}}\right\}
    \implies \mathcal{L}^{-1} \left\{\frac{1}{\left(s+\frac{1}{2}\right)^2+1}\ri  ght\}

    Apply the first translation theorem:

    \implies \mathcal{L}^{-1} \left\{\frac{1}{\left(s+\frac{1}{2}\right)^2+1}\ri  ght\}

    \implies f(t)=e^{-\frac{1}{2}t}\sin\left(t\right)

    \implies \color{red}\boxed{f(t)=\frac{\sin\left(t\right)}{\  sqrt{e^t}}}
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