Solving a homogeneous system of three ODEs with variable coefficients. I am interested in solving the following system of ODEs:
$$
\begin{pmatrix}
x'(t) \\ y'(t) \\z'(t)
\end{pmatrix} = a
\begin{pmatrix}
  0 & -B_2 & B_1 \sin \omega t \\
  B_2 & 0&  -B_1 \cos \omega t \\
  -B_1 \sin \omega t & B_1 \cos \omega t &  0
 \end{pmatrix}
\begin{pmatrix}
x(t) \\ y(t) \\z(t)
\end{pmatrix}
$$
either in MATLAB or analytically, but I have no idea how to do it. I'd appreciate any help on how to deal with this.
Is this even possible to solve analytically btw? The equation is of the form 
$$\frac{d{\bf v}(t)}{dt} = a \left( {\bf v}(t) \times {\bf B}(t) \right).$$
EDIT: So is there really no way to solve this in a smart way? Do I have to diagonalize? I don't know how to do it.

You can make a change of coordinate basis which diagonalises the $3\times 3$ matrix in the RHS. Then you will be able to solve this as three uncoupled DEs for your transformed basis. This will then give you a system of equations to solve for your original functions (or you may find that working in the transformed basis is more convenient anyway).
