In the paper, a generalized sub-equation method is presented to construct some exact analytical solutions of nonlinear partial differential equations. Making use of the method, we present rich exact analytical solutions of the one-dimensional nonlinear Schrödinger equation which describes the dynamics of solitons in Bose-Einstein condensates with time-dependent atomic scattering length in an expulsive parabolic potential. The solutions obtained include not only non-traveling wave and coefficient function's soliton solutions, but also Jacobi elliptic function solutions and Weierstrass elliptic function solutions. Some plots are given to demonstrate the properties of some exact solutions under the Feshbach-managed nonlinear coefficient and the hyperbolic secant function coefficient.