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Math Help - Laplace Transforms

  1. #1
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    Laplace Transforms

    Hi,

    For Question 2,

    I know you can do them by partial intergration but I'm not sure how to show them using the method it asks.

    For Question 4, I genuinely have no idea.

    Any help would be appreciated.

    Thanks

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  2. #2
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    Re: Laplace Transforms

    Quote Originally Posted by fourierT View Post
    Hi,

    For Question 2,

    I know you can do them by partial intergration but I'm not sure how to show them using the method it asks.

    For Question 4, I genuinely have no idea.

    Any help would be appreciated.

    Thanks

    By "using the definition" they mean

    $\large F(s) = \mathscr{L}\left \{f(t) \right \} = \displaystyle{\int_{-\infty}^{\infty}}f(t)e^{-st}~dt$

    and for 2(a) you would use "integration by parts" to evaluate this integral.

    for 2(b) just use their hint and do the integration.

    I should note that it looks like they implicitly mean f(t)=0 for t<0, otherwise these transforms won't converge.

    for 4) you should have read about using Laplace transforms to solve linear constant coefficient differential equations. This is a pretty straightforward example assuming H(t) stands for the Heaviside step function. Go re-read that section of your text. Or look at this.
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  3. #3
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    Re: Laplace Transforms

    Done the rest, still stuck on Question 4.

    Any further help would be great, thanks.
    Last edited by fourierT; March 22nd 2014 at 04:29 PM.
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  4. #4
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    Re: Laplace Transforms

    Quote Originally Posted by fourierT View Post
    Thanks,

    I'm slightly behind on the laplace stuff hence why I had a some trouble with these.

    I think I've done 2 a) and b) but still can't do 4).
    If we have two transform pairs $f(t) \overset{\mathscr{L}}{\Longleftrightarrow} F(s)$

    Then what does $\dfrac{d}{dt}f(t)$ correspond to in the s domain?
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  5. #5
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    Re: Laplace Transforms

    sF(s) - F(0) ?
    Last edited by fourierT; March 22nd 2014 at 05:22 PM.
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  6. #6
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    Re: Laplace Transforms

    No, that doesn't even make sense - derivative of F(s) with respect to t?

    The Laplace transform of f'(t) has an easy-to-write relationship to the Laplace transform of f(t). It should be in your textbook, or you can look at romsek's link.

    - Hollywood
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  7. #7
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    Re: Laplace Transforms

    Quote Originally Posted by fourierT View Post
    sF(s) - F(0) ?
    ok now apply this twice and take the Laplace transform of both sides of your differential equation and solve it in the s domain. Then transform it back to the t domain.
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