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Math Help - using the definition of Laplace Transform

  1. #1
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    using the definition of Laplace Transform

    Using the definition of Laplace transform and other theorems you find, determine: L{t(integral {0, t}) e^(-3u) du} Identify each theorem as you use it.

    If someone would help me with this I would really appreciate it. This problem will be on the final, so if you could clearly state which theorems need to be used for each step, I would really appreciate it.
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    Quote Originally Posted by Hollysti View Post
    Using the definition of Laplace transform and other theorems you find, determine: L{t(integral {0, t}) e^(-3u) du} Identify each theorem as you use it.

    If someone would help me with this I would really appreciate it. This problem will be on the final, so if you could clearly state which theorems need to be used for each step, I would really appreciate it.
    You want to find the Laplace transform of e^{-3u}?

    It seems you want to find the Laplace Transform of a Convolution? Right?
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    I don't know what a convolution is... And it is the Laplace of t times that integral...
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    Here.
    Attached Thumbnails Attached Thumbnails using the definition of Laplace Transform-picture13.gif  
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  5. #5
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    Quote Originally Posted by Hollysti View Post
    Using the definition of Laplace transform and other theorems you find, determine: L{t(integral {0, t}) e^(-3u) du} Identify each theorem as you use it.

    If someone would help me with this I would really appreciate it. This problem will be on the final, so if you could clearly state which theorems need to be used for each step, I would really appreciate it.
    You want the LT of:

    f(t) = t [integral_{0, t}) e^(-3u) du

    differentiate twice to eliminate the integral (we use the product rule and the
    fundamental theorem of calculus:

    f'(t) = integral_{0,t} e^{-3u} du + t e^{-3t}

    f''(u) = 2 e^{-3t} - 3 t e^{-3t)

    So Lf''(s) can be evaluated by table look-up, call it G(s), then:

    Lf''(s) = s^2 F(s) - s f(0) - f'(0)

    where F(s) = Lf(s), so:

    F(s) = [G(s) + s f(0) + f'(0)]/s^2

    Now f(0) = 0, and f'(0) = 0 so:

    F(s) = G(s)/s^2

    RonL
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