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Math Help - Laplace Transform from definition

  1. #1
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    Laplace Transform from definition

    Hi all,

    I was wondering if someone could possibly give me a step by step guide on how the laplace transfomation of f(t) = 3 + 2e^-kt is given from definition? ie using

    F(s) = integral of [ f(t) . e^-st dt ]?

    I can work out the actual transform but I dont follow how its done using the principal above.


    Thanks !
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  2. #2
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    Re: Laplace Transform from definition

    \displaystyle \begin{align*} F(s) &= \int_0^{\infty}{e^{-s\,t}\,f(t)\,dt} \\ &= \int_0^{\infty}{e^{-s\,t}\left(3 + 2e^{-k\,t}\right)dt} \\ &= \int_0^{\infty}{3e^{-s\,t} + 2e^{-\left(s + k\right)t}\,dt} \\ &= \lim_{\epsilon \to \infty}\left[ -\frac{3}{s}e^{-s\,t} - \frac{2}{s + k}e^{-\left(s + k\right)t} \right]_0^{\epsilon} \\ &= \lim_{\epsilon \to \infty}\left[ -\frac{3}{s}e^{-\epsilon \, s} - \frac{2}{s + k}e^{-\left( s + k \right)\epsilon } \right] - \left[ -\frac{3}{s}e^{-0t} - \frac{2}{s + k}e^{-\left(s + k\right)0} \right] \\ &= 0 - \left[ -\frac{3}{s} - \frac{2}{s + k}\right] \\ &= \frac{3}{s} + \frac{2}{s + k} \end{align*}
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    Re: Laplace Transform from definition

    thank you ! did you generate this from a website or is this your own working?
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  4. #4
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    Re: Laplace Transform from definition

    Quote Originally Posted by andyrobinson View Post
    thank you ! did you generate this from a website or is this your own working?
    I did it myself...
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    Re: Laplace Transform from definition

    Quote Originally Posted by andyrobinson View Post
    Hi all,

    I was wondering if someone could possibly give me a step by step guide on how the laplace transfomation of f(t) = 3 + 2e^-kt is given from definition? ie using

    F(s) = integral of [ f(t) . e^-st dt ]?

    I can work out the actual transform but I dont follow how its done using the principal above.


    Thanks !
    Any chance you're resitting 2nd year Mechanical Engineering maths at Glasgow Uni?
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