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

1. May 11, 2010

### jcx3

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. May 11, 2010

### Mute

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

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