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Math Help - Find a solution with Laplacetransform

  1. #1
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    Find a solution with Laplacetransform

    1. The problem statement, all variables and given/known data
    Determine the solution for
    y^{''}+81y=81U(t-\frac{\pi }{2})
    when \left\{y(0)=12,y'(0)=18\right\}


    U(t) is the unit step function




    3. The attempt at a solution

    Laplacetransforming :
    s^{2}Y(s)-sy(0)-y'(0)+81Y(s)=81e^{-\frac{\pi }{2}s}

    With given data the equation becomes

    s^{2}Y(s)-12s-18+81Y(s)=81e^{-\frac{\pi }{2}s}

    Solving Y(s)

    Y(s)=81e^{-\frac{\pi }{2}s}(\frac{1}{s^{2}+9^{2}})+12(\frac{s}{{s^{2}+9  ^{2}}})+18(\frac{1}{s^{2}+9^{2}})

    Transform again:
    y(t)=81U(t-\frac{\pi }{2})sin(9t)+12cos9t+18sin9t


    Problem: This is wrong but I dont know what am I doing wrong..Can someone tell me what Im doing wrong? Wolframalpha.com says that the solution should be: U(\frac{\pi }{2}-t)(sin(9t)-1)+sin(9t)+12cos(9t)+1

    But I dont understand how to get that answer.

    Thanks!!
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  2. #2
    Grand Panjandrum
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    Re: Find a solution with Laplacetransform

    Quote Originally Posted by Muffin View Post
    1. The problem statement, all variables and given/known data
    Determine the solution for
    y^{''}+81y=81U(t-\frac{\pi }{2})
    when \left\{y(0)=12,y'(0)=18\right\}


    U(t) is the unit step function




    3. The attempt at a solution

    Laplacetransforming :
    s^{2}Y(s)-sy(0)-y'(0)+81Y(s)=81e^{-\frac{\pi }{2}s}

    With given data the equation becomes

    s^{2}Y(s)-12s-18+81Y(s)=81e^{-\frac{\pi }{2}s}

    Solving Y(s)

    Y(s)=81e^{-\frac{\pi }{2}s}(\frac{1}{s^{2}+9^{2}})+12(\frac{s}{{s^{2}+9  ^{2}}})+18(\frac{1}{s^{2}+9^{2}})

    Transform again:
    y(t)=81U(t-\frac{\pi }{2})sin(9t)+12cos9t+18sin9t


    Problem: This is wrong but I dont know what am I doing wrong..Can someone tell me what Im doing wrong? Wolframalpha.com says that the solution should be: U(\frac{\pi }{2}-t)(sin(9t)-1)+sin(9t)+12cos(9t)+1

    But I dont understand how to get that answer.

    Thanks!!
    Your LT of the translated unit step is wrong

    \mathcal{L}[u(t-\tau)]=\frac{e^{-\tau s}}{s}

    CB
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  3. #3
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    Re: Find a solution with Laplacetransform

    Okey thx! But still.. Something is wrong. I dont get the right answer
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  4. #4
    Grand Panjandrum
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    Re: Find a solution with Laplacetransform

    Quote Originally Posted by Muffin View Post
    Okey thx! But still.. Something is wrong. I dont get the right answer
    So what do you now get?

    CB
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  5. #5
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    Re: Find a solution with Laplacetransform

    I get the same. When I inverstransform \frac{81e^{\frac{-\pi }{2}}}{s} I get 81U(t-\frac{\pi }{2})sin(9t)
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  6. #6
    Grand Panjandrum
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    Re: Find a solution with Laplacetransform

    Quote Originally Posted by Muffin View Post
    I get the same. When I inverstransform \frac{81e^{\frac{-\pi }{2}}}{s} I get 81U(t-\frac{\pi }{2})sin(9t)
    What is:

    \mathcal{L}^{-1}\left[ \frac{1}{s^2+a^2}\right] ?

    That may not get you all the way to the solution but it might help.

    CB
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