A Langevin differential equation which has a damping term and a control force both varying periodically at the same frequency, the equation being forced by an arbitrary function of time, is considered in this paper. The solution of the equation is reduced to that of a second-order linear differential equation in either the real or the complex domain ; no approximation is involved in this reduction, for it is exact. In particular, for the autonomous equation the exact solution shows that no limit cycles exist, contrary to the approximate results obtained by Ottoy for the equation whoso damping term fluctuates at twice the frequency of the control force. Next, a Langevin equation which is forced by a periodically fluctuating function of time (having arbitrary amplitude and frequency) is solved exactly when some of the parameters are interrelated in four distinct ways.