Abstract
In this paper, we analyze the damped Duffing equation by means of qualitative theory of planar systems. Under certain parametric choices, the global structure in the Poincaré phase plane of an equivalent two-dimensional autonomous system is plotted. Exact solutions are obtained by using the Lie symmetry and the coordinate transformation method, respectively. Applications of the second approach to some nonlinear evolution equations such as the two-dimensional dissipative Klein-Gordon equation are illustrated.