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An analytical solution to the nonlinear differential equation describing theequation of motion of a particle moving in an unforced physical system withlinear damping, governed by a cubic potential well, is presented in terms ofthe Jacobi elliptic functions. In the attractive region of the potential thesystem becomes an anharmonic damped oscillator, however with asymmetricdisplacement. An expression for the period of oscillation is derived, which fora nonlinear damped system is time dependent, and in particular it contains aquartic root of an exponentially decaying term in the denominator. Initiallythe period is longer as compared to that of a linear oscillator, howevergradually it decreases to that of a linear damped oscillator.  Transforming the undamped nonlinear differential equation into thedifferential equation describing orbital motion of planets, the perihelionadvance of Mercury can be estimated to 42.98 arcseconds/century, close topresent day observations of 43.1 arcseconds/century.  Some familiarity with the Jacobi elliptic functions is required, inparticular with respect to the differential behavior of these functions,however, they are standard functions of advanced mathematical computer algebratools. The expression derived for the solution to the nonlinear physicalsystem, and in particular the expression for the period of oscillation, isuseful for an accurate evaluation of experiments in introductory and advancedphysics labs, but also of interest for specialists working with nonlinearphenomena governed by the cubic potential well.

The Solution to the Differential Equation with Linear Damping describing a Physical Systems governed by a Cubic Energy Potential