Movement of charged particle in uniform magnetic field Being uniform, one can denote the magnetic field $\vec B = (B_x, B_y, B_z)$, where all $B_i$ are constants. I know that the particle inititial velocity   is $\vec v_0$ - I will use this later when integrating. The particle has charge $q$ and mass $m$.
Here, I can use Lorentz force ($\vec E = \vec 0$, because  $B$ is not time-dependent) combined with the Newton 2nd law:
$$m\dot{\vec v} = q(\vec v \times \vec B)$$
This results in three differential equations:
x'(t) = Cy-Bz, y' = -Cx + Az, z' = Bx - Ay
$$\frac{m}{q}\dot v_x = B_zv_y-B_yv_z$$
$$\frac{m}{q}\dot v_y = -B_zv_x+B_xv_z$$
$$\frac{m}{q}\dot v_z = B_yv_x-B_xv_y$$
However, the solution to this looks horrible as we can see on WolframAlpha. Is the colution really that complicated? I would expect a simple solution, because the problem looks pretty simple to me.
 A: The following may be useful.  You might find it helpful to simplify the problem first so you can see that the solutions directly make intuitive sense.
For example, take the $B$-field to only have a $B_{z}$ component.  In other words, set $A=B=0$ in your notation above.  When you rerun that in Wolfram, you should see a simpler solution for $x,y,z$ involving $\sin$ and $\cos$ and a constant (for z) as you might expect from the circular/spiral motion of a charged particle in a $B$-field.
Hopefully, this will get you going further.
I hope this helps.
A: Choose an other coordinate system with one of its axis, say the $z'$-axis, aligned to the
magnetic field. In this system the $z'$-component of the force is zero so the $z'$-component of the velocity is constant and the problem is somehow two dimensional.
Note that due to the "two-dimensionality" it would be convenient to use complex numbers like that
\begin{equation}
z\boldsymbol{=}x\boldsymbol{+}iy\,, \quad \dot{z}\boldsymbol{=}\upsilon_x\boldsymbol{+}i\upsilon_y\,, \quad \ddot{z}\boldsymbol{=}\dot{\upsilon}_x\boldsymbol{+}i\dot{\upsilon}_y\,, \quad \boldsymbol{-}i\dot{z}\boldsymbol{=}\upsilon_y\boldsymbol{-}i\upsilon_x
\tag{01}\label{01}
\end{equation}
