# Second Order Inhomogenous Differential Equation

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?

Thanks.

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