I think what joshi want is a solution to the above equations.
May be joshi was not using EJS or joshi did not know how to solve it.

It will be difficult to find a solution without know the form for the electric field E and magnetic field B.

It is relatively easy for inject a particle into a uniform magnetic field region.

You will find partial circular motion.
Assume the velocity $\vec{v}$ has component parallel to magnetic field $\vec{v_p}$  and perpendicular to component $\vec{v_n}$
i.e. $\vec{v}=\vec{v_p}+\vec{v_n}$
There is no force along the magnetic field, so it will move with constant speed along the direction of magnetic field with speed $v_p$
The velocity component perpendicular to magnetic field will cause particle to move in a circular orbit, which satusfy the following condition $m \frac{v_n^2}{r}= q v_n B$. i.e. $r= \frac{m v_n}{qB}$.

If the charged particle is in a uniform magnetic initially, then it will move with helix orbit.
However, if a charged is moving from a region where there is no magnetic field, and enter a region with uniform magnetic field.
The charged particle will exit the magnetic field within one resolution of circular motion.
Please check out [url=http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=179.0]Electron in magnetic field [/url]