Special analytical solutions of the damped-anharmonic-oscillator equation

The anharmonic-oscillator equation with dissipation has been widely used in various areas of science, and surprisingly little has been known about its analytical solutions. In this paper, we survey the known classes of solutions and the methods employed to obtain them including the Painlev\'e Test and its results. We propose several approaches, one of which is based on compatibility with a Bernoulli equation. Another method investigated explores the many solutions that take the form of a rational fraction of linear combinations of exponentials. In addition to elementary function solutions, those involving elliptic functions are examined. The way the restriction on the parameters of the equation may be relaxed somewhat is demonstrated, leading to an extension of elliptic-type solutions.