A set of coupled ODE I've recently encountered this set of coupled ODE,
$\partial_x\alpha(x)-iA(x)\beta(x)+B(x)\beta(x)=iC(x)\alpha(x),\\
\partial_x\beta(x)+iA(x)\alpha(x)+B(x)\alpha(x)=-iC(x)\beta(x),$
where $\alpha(x)$, $\beta(x)$, $A(x)$, $B(x)$ and $C(x)$ are real functions of $x$. This set of coupled ODE was arisen from solving a 2x2 matrix eigenvalue problem. Naively this should not be hard, but I got stucked for a while... 
I tried to use the matrix method $\mathbb{x}'=\mathbb{A}\mathbb{x}$ to solve it, but faced two difficulties: the complex entries and all matrix elements are functions which makes it hard to determine the eigenvalues. 
As a remark, I already generalized from the set of coupled ODE I'm dealing with and hope to find a general form of solutions by replacing original functions with $A(x), B(x), C(x)$.
*I consulted with a few applied mathematicians and physicists about this problem and they concluded that there is no solution of analytic form (product of elementary functions) which I highly doubted. I believe this can be solved analytically and systematically.
So, here I am. Please point out my flaws and enlight me.
Thanks in advanced. 
 A: This system can be written as
\begin{equation}
\frac{d}{dx}
\begin{pmatrix}
\alpha\\
\beta
\end{pmatrix}
\;=\;
{\bf M}(x)
\begin{pmatrix}
\alpha\\
\beta
\end{pmatrix}\, ,
\end{equation}
where the complex matrix ${\bf M}(x)$ is
\begin{equation}
{\bf M}(x)
\;=\;
\begin{pmatrix}
i C(x) & i A(x) - B(x)\\
-iA(x)-B(x) & -i C(x)
\end{pmatrix}\, .
\end{equation}
If the initial conditions at $x = x_o$ are $\alpha_o$ and $\beta_o$, then the general solution to this set of ODEs is
\begin{equation}
\begin{pmatrix}
\alpha\\
\beta
\end{pmatrix}
\;=\;
\exp \left[\int_{x_o}^x\, dx'\, {\bf M}(x')
\right]\, \begin{pmatrix}
\alpha_o\\
\beta_o
\end{pmatrix}\, .\qquad\qquad\qquad
\text{(Not generally valid!  See below.)}
\end{equation}
Here the exponential is meant to be understood in terms of matrix exponentiation.
Without more knowledge about the nature of the functions $A$, $B$, and $C$, I don't believe much else can be said about this solution.
ETA: Note: It's been pointed out (see comments below) that this solution is only valid when $\int_{x_o}^x\, dx'\, {\bf M}(x')$ commutes with ${\bf M}(x)$, which will in general not be true. At the request of the original poster, I'm leaving this "answer" up.
By the way, I find that for fixed $x$, the eigenvalues of the matrix ${\bf M}(x)$ above are
\begin{equation}
\lambda_{\pm}(x)\;=\;
\pm\sqrt{A(x)^2 + B(x)^2 - C(x)^2}\, .
\end{equation}
The corresponding eigenvectors can be also be found in the usual way.
