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Green's Function for Third Order ODE

  1. Dec 5, 2013 #1
    1. The problem statement, all variables and given/known data
    A dynamical system has a response, y(t), to a driving force, f(t), that satisfies a differential equation involving a third time derivative:

    [itex]\frac{d^{3}y}{dt^{3}} = f(t)[/itex]

    Obtain the solution to the homogeneous equation, and use this to derive the causal Green's function for this system, G(t;τ). [hint: which order of derivative has a discontinuity at t = τ?]

    2. The attempt at a solution

    I've obtained a solution to the homogeneous equation [itex]\frac{d^{3}y}{dt^{3}} = 0 [/itex] by integrating 3 times with respect to t giving [itex]y(t) = \frac{1}{2}At^{2} + Bt + C[/itex].

    Since I'm looking for a causal Green's function I know for t<τ G(t;τ) = 0.

    Taking the advise of the hint I have tried to find which order of derivative has a discontinuity at t=τ. First replacing the driving force, f(t), with a delta function, δ(t-τ), I get

    [itex]\frac{d^{3}y}{dt^{3}} = \delta(t-τ)[/itex]

    then integrating over the interval [τ-ε, τ+ε] and letting ε tend to 0, I conclude the second derivative changes discontinuously by 1.

    Is this correct or have I missed something in determining where the discontinuity lies?
    Thanks.
     
  2. jcsd
  3. Dec 5, 2013 #2

    vela

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    Looks fine.
     
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