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Nonhomogeneous ODE with Dirac delta

  1. Apr 25, 2008 #1
    Trying to solve the ODE
    [tex]mx''(t) + bx'(t) + kx(t) = F(t)[/tex]
    with [tex]m[/tex] measured in [tex]Kg[/tex], [tex]b[/tex] in [tex]Kg/s[/tex] and [tex]Kg/s^2[/tex], [tex]F(t)[/tex] in [tex]Kgm/s^2[/tex] and [tex]x(t)[/tex] in [tex] m[/tex] with initial conditions [tex]x(0) = 0[/tex] and [tex]x'(0) = 0[/tex], i got the following Green's function

    [tex]G(t,t') = \frac{1}{m\omega} e^{-\omega_1(t-t')}\sinh\left[\omega(t-t')\right][/tex]
    for [tex]t \geq t'[/tex], with
    [tex]\omega = \sqrt{\frac{k}{m}}[/tex]

    Adding up all the responses to unit impulses i got

    [tex]x(t) = \int_0^t F(t')G(t,t')dt'[/tex]

    So far, everything was ok.

    But when i tried to get the response of the ODE for a unit impulse of amplitude [tex]f_0[/tex] at [tex]t = t_0[/tex] ([tex]F(t) = f_0\delta(t-t_0)[/tex]) i integrated the equation above but the dimension of the expression i got for [tex]x(t)[/tex] is [tex]m/s[/tex] instead of [tex]m[/tex].

    If [tex]F(t)[/tex] is any other function it works fine, but with [tex]F(t) = f_0\delta(t-t_0)[/tex] I get this dimensional error.
    Can someone help me, please?
  2. jcsd
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