Exact solutions to a family of position-dependent mass damped oscillators from variational λ-symmetries

A wide family of position-dependent mass damped oscillators affected by an external potential is investigated. First, a Lagrangian formulation is introduced for the corresponding problem. The Lagrangian function is time-dependent, and the problem cannot be approached with classical procedures because it lacks variational symmetries. Therefore, the variational lambda-symmetry method is applied to find exact solutions. Variational lambda-symmetries are determined for a family of potential functions, which lead to a one-parameter family of exact solutions. The results are applied to particular examples corresponding to some interesting mass functions reported in the previous literature.