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Math Help - Laplace of the first and second derivatives

  1. #1
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    Laplace of the first and second derivatives

    It's been a long time since I last used Laplace so please forgive me if this is a dumb question.

    Assuming I take the first and second derivatives of a function should the Laplace transforms of those derivatives be the same?
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  2. #2
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    If you have a function f(t), with corresponding Laplace transform F(s), then the Laplace transform of f'(t) is s F(s)-f(0), where the upper- and lower-case letters are very carefully typed. Similarly, the second derivative of f(t) would have the Laplace transform s^{2}F(s)-sf(0)-f'(0). So you gain two things by Laplace transformation: differentiation becomes multiplication, and the initial conditions are automatically included in your answer! Of course, not all functions even have a Laplace transform, so this method can only get you so far.
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