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Math Help - Laplace Transformation

  1. #1
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    Laplace Transform

    I'm a little a bit confused about the following exercise because of the two segments of the function. How can we find the Laplace transformation of this function

    f(t) = \begin {cases} t , 0\le t < 4 \\ 5 , t\ge 4\end {cases}

    Thanks in advance!
    Last edited by ypatia; March 25th 2009 at 02:01 AM. Reason: Change the title
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  2. #2
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    Just use the definition of the Laplace transform:
    F(s)=\int_0^{\infty}e^{-st}f(t)dt
    =\int_0^{4}e^{-st}\cdot{t}\,dt+\int_{4}^{\infty}e^{-st}\cdot{5}\,dt
    You should be able to figure out those integrals.

    --Kevin C.
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  3. #3
    MHF Contributor chisigma's Avatar
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    The simplest way is to solve the integral...

    \int_{0}^{\infty} f(t)\cdot e^{- s\cdot t}\cdot dt = \int_{0}^{4} t\cdot e^{- s\cdot t}\cdot dt + \int_{4}^{\infty} 5\cdot e^{- s\cdot t}\cdot dt =

    = - \frac{1}{s}\cdot |t\cdot e^{- s\cdot t}|_{0}^{4} - \frac{1}{s^{2}}\cdot |e^{- s\cdot t}|_{0}^{4} - \frac{5}{s}\cdot |e^{- s\cdot t}|_{4}^{\infty} = \frac{1- (1-s)\cdot e^{-4\cdot s}}{s^{2}}

    Kind regards

    \chi \sigma
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