On integrable non–autonomous Liénard–type equations

We study a family of nonautonomous generalized Liénard-type equations. We consider the equivalence problem via the generalized Sundman transformations between this family of equations and type-I Painlevé–Gambier equations. As a result, we find four criteria of equivalence, which give four integrable families of Liénard-type equations. We demonstrate that these criteria can be used to construct general traveling-wave and stationary solutions of certain classes of diffusion–convection equations. We also illustrate our results with several other examples of integrable nonautonomous Liénard-type equations.