= Solution
Assume $E,B>0$ for the directions and parametrizations; the boost formulas retain their signed meaning for reversed fields. A <Lorentz boost> with <velocity> $w\hat{\mathbf z}$ along $\mathbf E\times\mathbf B$ transforms the transverse <electric field> and <magnetic field> as
$$
E'_x=\gamma_w(E-wB),\qquad B'_y=\gamma_w(B-wE/c^2).
$$
In the electric-dominated case choose
$$
\boxed{\mathbf w=\frac{c^2\mathbf E\times\mathbf B}{E^2}
=\frac{c^2B}{E}\hat{\mathbf z},\qquad
\mathbf B'=0,\qquad E'_x=E\sqrt{1-c^2B^2/E^2}.}
$$
The inequality ensures $|w|<c$, so this is an <inertial frame>. The <field-removing boosts for perpendicular electric and magnetic fields> are governed by the <electromagnetic field invariants> $\mathbf E\cdot\mathbf B=0$ and $E^2-c^2B^2>0$ explain why a purely electric <inertial frame> exists and a purely magnetic one does not.
The <electron> is \b[not initially at rest] in this convenient <inertial frame>: its initial <velocity> is $-w\hat{\mathbf z}$ and its initial <three-momentum> is $p'_z=-\gamma_wmw$. Thereafter $p'_x=-eE't'$, while $p'_z$ stays constant. The <electric field> accelerates it toward $-x$; as its <relativistic energy> increases, its negative transverse <velocity> decreases in magnitude. Defining $a=eE'/m>0$ and $s=a\tau/c$ gives a useful parametric <world line>:
$$
t'=\frac{\gamma_wc}{a}\sinh s,\qquad
x'=-\frac{\gamma_wc^2}{a}(\cosh s-1),\qquad
z'=-\frac{\gamma_wwc}{a}s.
$$
Here $\tau$ is the <proper time>. These formulas follow by integrating $U'^0=\gamma_wc\cosh s$, $U'^x=-\gamma_wc\sinh s$, and constant $U'^z=-\gamma_ww$. Its projection onto the $t',x'$ plane is a <hyperbola>, but the spatial trajectory includes the initial transverse motion and is not a straight line.
Transforming back with $z=\gamma_w(z'+wt')$, $t=\gamma_w(t'+wz'/c^2)$ gives
$$
x=x',\qquad z=\frac{\gamma_w^2wc}{a}(\sinh s-s),\qquad
t=\frac{\gamma_w^2c}{a}(\sinh s-w^2s/c^2).
$$
The laboratory trajectory starts from rest toward $-x$, then bends toward $+z$. It is unbounded in $-x$ and ultimately has $u_z\to w=c^2B/E$ and $u_x\to-c/\gamma_w$. Thus the late <velocity> has magnitude $c$ as a limit, consistent with the <relativistic energy-momentum relation>, rather than undergoing a closed <cyclotron orbit>.
\Image[/past-exam-of-the-symmetry-and-relativity-course-of-the-university-of-oxford/2021-crossed-field-trajectories.png]
{title=Electron trajectories in electric-dominated and magnetic-dominated crossed fields}
{description=Original parametric spatial trajectories in the convenient <inertial frame> and laboratory, with dimensionless units $c=m=e=1$. Arrows show increasing time. The initial transverse motion in the purely electric frame and the laboratory cusps in the purely magnetic case are visible.}
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