Trajectory in magnetic undulator

[Franky4]

Homework Statement

I am asked to find r(t) for a charged q particle in an magnetic undulator. Wrote down these equations:
x'' = w* y' *cos(a*x) (1)
y'' = -w* x' *cos(a*x) (2)
z'' = 0 (3)

r(0) = (x0, y0, z0); r(0)' = (x0', y0', z0').

Homework Equations

Not sure how to go on solving these.

The Attempt at a Solution

z(t) is obvious. I am able to integrate (2) once to find y' = -w/a *sin(a*x) + C. Plugging it into (1) doesn't seem to do any progress, since I get x'' = - w^2 /a *sin(a*x)*cos(a*x) + C*w*cos(a*x). Because particle is in magnetic field, it's known that sqrt(x'^2 + y'^2 + z'^2) = constant from r(0)', but not sure how to use it to my advantage.

 

[mfb]

A typical undulator would allow some approximations that make the equations easier (e. g. "x' does not change much").
A Fourier transformation could give interesting results (the path is like a sine-curve or a bit similar to a circle, but certainly periodic), but I don't know if it works.