An analytical solution to the differential equation describing the Duffing oscillator with damping
is presented. The damping term of the differential equation and the initial conditions satisfy an
algebraic equation, and thus the solution is specific for this type of damping. The nonlinear term
of the differential equation is allowed to be considerable compared to the linear term. The solution
is expressed in terms of the Jacobi elliptic functions by including a parameter-dependent elliptic
modulus. The analytical solution is compared to the numerical solution, and the agreement is found
to be very good. It is established that the period of oscillation is shorter compared to that of a
linearized model but increasing with time and asymptotically approaching the period of oscillation
of the linear damped model. An explicit expression for the period of oscillation has been derived,
and it is found to be very accurate.