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(d^2R/dt^2 ) + (dR/dt) x B = 0

where the capitals are vectors (sorry i suck at latex). R is a position vector in x-y plane and B is in the z-direction.

do i split this into equations for x and y directions separately and solve them? for x-direction i would get

d^2x/dt^2 + (dy/dt)|B| = 0

but how do i solve this? I'm pretty sure all the DEs I've ever solved had, for example, d^2x/dt^2 and dx/dt in them, but not d^2x/dt^2 and dy/dt.

or is there a quick way of doing it without having to expand the cross product?

any help would be well appreciated!