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.

I don't have Matlab, but here's the Mathematica code to do it:
solx =With[{a=1,\[Omega]=1,b2=1,b1=2},
           NDSolve[{x'[t]==-a*b2*y[t]+a*b1*Sin[\[Omega]*t]*z[t],
                    y'[t]==a*b2*x[t]-a*b1*Cos[\[Omega]*t]*z[t],
                    z'[t]==-a*b1*Sin[\[Omega]*t]*x[t]+a*b1*Cos[\[Omega]*t]*y[t],
                    x[0]==1,y[0]==0,z[0]==0},
                    {x,y,z},{t,0,20}]]
ParametricPlot3D[{Evaluate[{x[t], y[t], z[t]} /. solx]}, {t, 0, 20}]

I plotted this for a few $\mathbf{v}(0) = (0,0,1), (0,1,0)$, and $(1,0,0)$, giving the following:
