The study concerns the application of a novel idea used to find analytical solutions of a nonlinear boundary value problem which arises in non-Newtonian channel flow. The velocity profile of a viscoelastic fluid between two planes is obtained with both slip and no-slip boundary conditions modeled by an Oldroyd 6 constant fluid. We present analytical solutions on two classical problems- the Poiseuille flow and the Couette flow- of a viscoelastic fluid between two parallel planes. We solved this equation by reducing it to a cubic equation in the derivative and the unknown. We used the cubic formula to solve this equation. Finally, we integrated this expression with binomial expansion to obtain a purely analytic series solution to this problem.