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Math Help - Laplace transform for IVP

  1. #1
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    Question Laplace transform for IVP

    Hey guys, I'm wondering if anyone can show me the steps for this problem. The question is below, I have be able to solved part a and b, but for part c, I know how to get the first two terms but I am not sure how to get the answer for the last term.
    thanks

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  2. #2
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    Quote Originally Posted by hazeleyes View Post
    Hey guys, I'm wondering if anyone can show me the steps for this problem. The question is below, I have be able to solved part a and b, but for part c, I know how to get the first two terms but I am not sure how to get the answer for the last term.
    thanks

    If you take the Laplace transform we get

    For the first part just evaluate this integral

    \int_{0}^{\infty}e^{at}\cdot e^{-st}dt = \int_{0}^{\infty}e^{-t(s-a)}dt

    s^2X(s)-s(x(0)-x'(0)-a^2X(s)=F(s) \iff X(s)=\frac{s-1}{xs^2-a^2}+F(s)\cdot \frac{1}{x^2-a^2}

    Now use the convolutions theorem to finish the job remember that

    \mathcal{L}^{-1}\{F(s)G(s) \}=\int_{0}^{t}f(t-u)g(u)du

    Can you finish from here?
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  3. #3
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    Thanks for your replied. I'm sorry, is it ok if you can show to the step by using this theorem.
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  4. #4
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    Quote Originally Posted by hazeleyes View Post
    Thanks for your replied. I'm sorry, is it ok if you can show to the step by using this theorem.
    So in your problem you have

    \mathcal{L}^{-1}\{F(s)\}=f(t)

    and

    \mathcal{L}^{-1}\{G(s)\}=\mathcal{L}^{-1}\{\frac{1}{s^2-a^2}\}=\frac{1}{a}\sinh(at)

    Now just plug these into the convolution theorem.
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