How to solve a tensor differential equation?

Essentially, How does one solve the tensorial differential equation $$\frac{dx^a}{d\tau}=A^a{}_bx^b$$

where $x^a$ is a 4-vector and $A^a{}_b$ is a $(1,1)$ tensor.

The original Problem

How does one solve the tensor differential equation for the relativistic motion of a partilcle of charge $e$ and mass $m$, with 4-momentum $p^a$ and electromagnetic field tensor $F_{ab}$ of a constant magetic field $\vec B$ perpendicular to the plane of motion. $$\frac{dp^a}{d\tau}=\frac{e}{m}F^a{}_bp^b$$ ? Let the the initial condition be $$p^a=(E_0 ,\vec 0)$$

I can see that the differential equation resembles that of a SHM equation or a cosh, sinh one if it's a scalar equation. However, I don't know how to deal with a tensor equation. Could anyone please explain? Thank you.

• What if you replace tensor by matrix ? – Damien L May 9 '13 at 7:56