Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    Mute

    User Avatar
    Homework Helper

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Exp(sin(t)) integral difficulty for 1st order ODE
Loading...