I have run into an issue trying to solve this second order differential equation

$ r''(t) - i r'(t) = -i\gamma[-\frac{1}{2} + \frac{1}{1+e^{-\alpha t}}], $

where $\alpha$ and $\gamma$ are real constants. I have tried to take the Laplace transform, and I have also tried to use the method of variation of parameters, where I have pulled two linearly independent basis functions from the homogeneous solution

$ r_0(t) = -i\dot{r}_ce^{it} +r_c, $

where $\dot{r}_c$ and $r_c$ are constants of integration.

I can not seem to easily solve for the particular solution, $r_p(t)$, because of the presence of the logistic inhomogeneity. Is it not generally possible to solve this SODE with a closed form solution in elementary functions?


  • $\begingroup$ You might try seeing how far variation of parameters gets you $\endgroup$ – DaveNine Sep 3 '15 at 23:38
  • $\begingroup$ In this case I eventually get to an integral that has no closed form solution in elementary functions. $\int{\exp(it)/(1-\exp(-\alpha t)) dt}$ $\endgroup$ – Loonuh Sep 3 '15 at 23:50

The solution can be expressed on the form of the antiderivative of a particular hypergeometric function. The most likely, there is no simpler closed form in terms of a finite number of elementary and standard special functions.

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