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Laplace Transform of Step Function

  1. Jun 21, 2012 #1

    ElijahRockers

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

    Solve

    y'' + y = f(t), y(0)=0, y'(0)=1,

    f(t)=
    (0 for 0<t<pi)
    (1 for pi<t<2pi)
    (0 for t>2pi)

    3. The attempt at a solution

    y'' + y = upi(t)-u2pi(t)

    s2L{y} -sy(0) -y'(0) +L{y} = L{upi(t)} -Lu2pi(t)}

    L{y}(s2+1) -1 = (e-pi*s/s) -(e-2pi*s/s)

    L{y} = (e-pi*s/s(s2+1)) -(e-2pi*s/s(s2+1)) +1/(s2+1)

    This is where I get stuck... I'm assuming that I can factor out the e terms separately, then use decomposition of partial fractions to separate 1/s(s2+1), but when I do that I get meaningless values for A and B.

    1/s(s2+1) = A/s + B/s2+1

    1= A(s2+1) +Bs

    1 = As2 +Bs +A

    From that I can infer that A = 1, but also that A=0, and B=0.

    What am I doing wrong?
     
  2. jcsd
  3. Jun 21, 2012 #2

    vela

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    The partial fraction expansion should be
    $$\frac{1}{s(s^2+1)} = \frac{A}{s} + \frac{Bs+C}{s^2+1}$$ because the second term has a quadratic in the denominator.
     
  4. Jun 21, 2012 #3

    ElijahRockers

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    Ah, thank you, I think I got it.

    y = upi(t) -upi(t)cos(t-pi) -u2pi(t) +u2pi(t)cos(t-2pi) +sin(t)
     
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