1. The problem statement, all variables and given/known data Consider a charged particle entering a region of uniform magnetic field B - for example the earth's field. Determine it's subsequent motion when the y-axis is parallel with the magnetic field 2. Relevant equations F=qv x B = ma vector components of velocity, acceleration, and magnetic field 3. The attempt at a solution This isn't homework, it's an example from the book, but they skip many gory details that I can't seem to figure out, and the way we did it in class makes even less sense. To cut a long story short, eventually I get two equations: z'''=-α2z' and x'''=-α2x' Then the book says they use the technique's used in appendix C, which are the use of Auxiliary functions (or characteristic) and homogeneous equations, which we aren't using in class apparently, I'd like to know how they are used and the examples given in the back make sense, but they are all 2nd degree functions, not 3rd like mine. My attempt with the equations as they are: r3+α2r=0 which I can see 3 solutions from: r=0, r=±iα the second solution will yield the cos and sin functions I desire, but I'm missing a "t" that needs to be with the alpha in the trig functions and I have no idea how to get it. x=Acos(αt) + Bsin(αt) + x0 z=A'cos(αt) + Bsin(αt) + z0 These two are the end results in the book I feel as if though I'm supposed to integrate the functions first which will give me the initial position constants, but I don't know where the "t" comes from.