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Exp(sin(t)) integral difficulty for 1st order ODE

  1. May 11, 2010 #1
    A model of tank fluid temperature (fixed mass inside) based on the mixing of two streams, one hot and one cold, each with fixed temperature, the hot one a fixed flowrate and the cold one an oscillatory flowrate can be expressed as:

    du/dt + [c1 + c2*(1+cos(t))]*u = c1

    The solution can be solved easily with an integrating factor and written in terms of an integral. I am trying to derive an expression for the AMPLITUDE of the asymptotic oscillation of u(t)which comes down to reducing the integral:

    Int[0,t] { exp[ - t' - k1*sin(k2*(t'-t)) ] } dt'

    I saw a reference suggesting this could be expressed as a Bessel function, but I can't see how, and I'm having a very hard time finding ANY references to integrals of exponentials of trigonometric functions. ANY INSIGHT WOULD BE GREATLY APPRECIATED!
  2. jcsd
  3. May 11, 2010 #2


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    If you check out Wikipedia's Bessel function page, you can find the identity

    [tex]e^{z\cos\theta} = I_0(z) + 2\sum_{n=1}^\infty I_n(z) \cos(n\theta)[/tex]
    where [itex]I_n(z)[/itex] is the nth order modified Bessel function of the first kind. For your case, set [itex]\theta = \phi + \pi/2[/itex] to get it in terms of sine.
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