Affine evolution of chain end-to-end vectors distribution function is derived analytically for non-linear polymer liquids subjected to uniaxial elongational flow, controlled by time-evolution of chain deformation coefficients. Peterlin approximation for non-Gaussian chain elasticity is applied, with Pade approximation for the inverse Langevin function. The approach enables calculations of transient molecular deformation coefficients in entire range of elongation rates and times. Equations controlling time evolution of the molecular deformation coefficients in elongational flow are solved analytically with an assumption of dominating elongational component. The approach allows to decouple evolution equations and obtain an approximate closed form analytical formula describing time evolution of the molecular deformation with high accuracy, in particular at higher elongation rates, above the Gaussian limit. Predictions of the analytical formula are compared with numerical computations to evaluate the approximation and ranges of its validity. The analytical formula enables predicting evolution of average functions in non-linear systems, such as free energy, tensile stress, molecular orientation, etc. The formula is used to discuss molecular vs. macroscopic deformation in wide range of elongation rates and times, as well as evolution of stress, axial orientation factor, apparent elongational viscosity. © 2005 Elsevier Ltd. All rights reserved.