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Inverse Laplace Transform

  1. Dec 29, 2007 #1
    1. Find [tex]v(t)[/tex] if [tex]V(s)=\frac{2s}{(s^{2}+4)^{2}}[/tex]

    Ans: [tex]v(t)=\frac{1}{2}tsin2tu(t)[/tex]

    2. Relevant equations:

    [tex]V(s)=\frac{a_{n}}{(s-p)^{n}}+\frac{a_{n-1}}{(s-p)^{n-1}}+\cdots+\frac{a_{1}}{(s-p)}[/tex]
    [tex]a_{n-k}=\frac{1}{k!}\frac{d^{k}}{ds^{k}}[(s-p)^{n}V(s)]_{s=p}[/tex]

    3. Attempt at a solution:

    [tex]V(s)=\frac{2s}{(s^{2}+4)^{2}}[/tex]

    [tex]V(s)=\frac{2s}{(s^{2}+4)^{2}}=\frac{A}{(s^{2}+4)^{2}}+\frac{B}{(s^{2}+4)}[/tex]

    [tex]A=\left[2s-B(s^{2}+4)\right]_{s=2i}[/tex]

    [tex]A=4i[/tex]

    [tex]B=\frac{d}{ds}\left[2s-B(s^{2}+4)\right]_{s=2i}[/tex]

    [tex]B=2[/tex]

    [tex]V(s)=\frac{4i}{(s^{2}+4)^{2}}+\frac{2}{(s^{2}+4)}[/tex]

    I am not sure what to do with the imaginary term, but it does not translate to 1/2t, which is what is required for the answer.

    [tex]?+sin2tu(t)[/tex]


    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Dec 29, 2007 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    No. Since the denominator, [itex]s^2+ 4[/itex] is quadratic you need
    [tex]\frac{2s}{(s^2+4)^2}= \frac{Ax+ B}{(x^2+4)^2}+ \frac{Cx+ D}{x^2+4}[/tex]
     
  4. Dec 29, 2007 #3
    Thanks, I'll attempt again.
     
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