  We have carried out an approximate analytical solution to precisely consider
the influence of magnetic field on the transverse oscillation of particles in
cyclotron. The differential equations of transverse oscillation are solved from
the Lindstedt-Poincare method. After careful deduction, the accurate first
order analytic solutions are obtained. The analytical solutions are applied to
the magnetic field, comes from an isochronous cyclotron with four spiral
sectors, the accuracy of these analytical solutions is verified and confirmed
from the comparison of numerical method. Finally, we discussed the transverse
oscillation at v0=N/2 , using the same analytical solution.
