In this work, we use the same strategy studied in our previous work [J. I. Jiménez-Aquino and M. Romero-Bastida, Phys. Rev. E 74, 041117 (2006)] to solve exactly the Fokker-Planck (FP) and Fokker-Planck-Kramers (FPK) equations of a charged Brownian particle in a fluid (a heavy ion in a light gas) under the influence of external fields: a constant magnetic field and, in general, time-varying mechanical and electric fields. In our proposal, a time-dependent rotation matrix is introduced to transform the Langevin equation in the phase-space $\left(\mathbf{r},\mathbf{u}\right)$ to a new space $\left({\mathbf{r}}^{\prime },{\mathbf{u}}^{\prime }\right)$. As a result, the transformed Langevin equations are very similar to those of ordinary Brownian motion in the presence of those time-varying external forces only, without the magnetic field; therefore, the associated FP and FPK equations can easily be solved in those transformed spaces. To solve these equations, we use the methods of solution developed by Chandrasekhar in the field-free case of ordinary Brownian motion. We also calculate a more general transition probability density in the velocity space by assuming an initial heavy-ion Maxwellian distribution at a temperature generally different from that corresponding to equilibrium, the same as that used by Ferrari [Physica A 163, 596 (1990)].

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