Decoupling coupled differential equations with time dependent coefficients Consider the following system of coupled differential equation.
$$\left[ \begin{array}{c} \frac{dc_1}{dt} \\ \frac{dc_2}{dt} \end{array} \right] = \begin{bmatrix} -B & -V(t) \\ -V(t) & B \end{bmatrix} \times \left[ \begin{array}{c} c_1 \\ c_2 \end{array} \right]$$
I tried diagonalyzng the matrix using the eigenvectors of the coefficient matrix. But, since the matrix is time dependent so are its eigenvectors. So, how can one decouple this system? 
 A: You can't. As you say, the eigenvectors of this matrix are time dependent, so it's no use to make linear combinations of $c_1$ and $c_2$ using these eigenvectors, because the time derivative will introduce extra terms by the product rule.
What you can do, is try to write $c_1(t) = \alpha(t) x(t)$ and $c_2(t) = \beta(t) y(t)$, and try to choose $\alpha(t)$ and $\beta(t)$ such that the system simplifies. As it turns out, if you choose $\alpha(t) = e^{-B t}$ and $\beta(t) = e^{B t}$, you obtain
\begin{equation}
 \begin{pmatrix} \frac{\text{d} x}{\text{d} t} \\ \frac{\text{d} y}{\text{d} t} \end{pmatrix} = \begin{pmatrix} 0 & -e^{2 B t} V(t) \\ - e^{-2 B t} V(t) & 0 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}.
\end{equation}
This is equivalent to the second order ODE
\begin{equation}
 \frac{\text{d}^2 x}{\text{d} t^2} - \left(2 B + \frac{V'(t)}{V(t)}\right)\frac{\text{d} x}{\text{d} t} - V(t)^2 x = 0,
\end{equation}
which is of Sturm-Liouville type. For general $V(t)$, there are only general results for the solutions of such equations (see e.g. Titchmarsh, Eigenfunction Expansions). If $V(t)$ is periodic in time though, you can invoke Floquet theory.
