The coupled differential equations of interest are
\begin{align} i\dot{C}_{+}(t)=\cos(\omega t)e^{2 i \alpha t} C_{-}(t) \end{align}
and
\begin{align} i\dot{C}_{-}(t)=\cos(\omega t)e^{-2 i \alpha t} C_{+}(t) \end{align}
Solving the first equation above for $C_{-}(t)$ differentiating it, plugging it into the second equation above, and simplifying we have
\begin{align} \ddot{C}_{+}(t) = -\cos^2(\omega t)C_{+}(t) + [2i\alpha-\omega\tan(\omega t)]\dot{C}_{+}(t) \end{align}
The initial conditions of interest are $C_{+}(0)=1$ and $C_{-}(0)=0 \implies \dot{C}_{+}(0)=0$ and I wish to have an analytic solution for t>0
For $\alpha \in Reals$ and $\omega \in Reals$
Mathematica can't seem to solve this problem analytically using DSolve and numerically using NDSolve it can only find the solution for a finite time after which it has no solution because "the step size is effectively 0".
I wish to have an analytic solution. I have tried Fourier transforming the equation in order to solve it with little luck but that still may be the way to go.
Thanks for any and all help!