We can choose to align the z axis with the magnetic field B. Then:

, ,

So:

Solving the third equation by integrating twice with respect to t: where k is a constant.

Now if you differentiate the second equation and then substitute for from the first then you will get:

which has the general solution:

So

Now using equation 1

So

Now using , , we can write:

This is the equation of a helix.