Invariants of a family of third-order ordinary differential equations

We consider a family of third-order ordinary differential equations (ODEs) with the cubic nonlinearity in second-order derivative, which is closed with respect to contact transformations. Using Lie's infinitesimal method we construct the basis of differential invariants for this family of equations and the operators of invariant differentiation. Invariants provide a simple way of finding the equations, which may be equivalent to a given nonlinear ODE and the transformation connecting two equivalent equations. Some examples are given to illustrate the results obtained in the paper.