We can rewrite Eq. (13) in the form where By introducing a new independent variable z = ∫ F$_{1}$ ( H ) dH , we obtain with g ( z ) is defined parametrically as, As shown in Appendix B, g ( z ) can be approximated as a simple function of z where C is a constant. We proceed with this approximation to get solvable differential equa- tions. Keeping the parametric solution of g ( z ), Eq. (50), results in much more complicated differential equations. This would be the subject of a future work. From the definitions of Ω and z we have Analogous to the solution of reduced Abel type canonical equation, (see Appendix C) we obtain the relation Ω = z/ P . Therefore, from Eqs. (18) and (19) we obtain the following first order differential equation for Hubble parameter H , P ˙ H = − 3[1 + (1 − r ) γ ] 2(1 − r ) H 2 + 1 αr H (21) y dy dx − y = ax (20) Ω = H 1+ r ˙ H, (18) z = H 2+ r ( − 3[1 + (1 − r ) γ ] H 2(1 r ) + 1 αr ) , (19) 7 g ( z ) ≈ C z, (17) g ( z ) = F$_{0}$ F$_{1}$ . (16) Ω d Ω dz − Ω = g ( z ) , (15) Ω d Ω dH = F$_{1}$ ( H )Ω + F$_{0}$ ( H ) , (14) Ω = u E = u exp ( − ∫ 1 + r H dH ) , F$_{1}$ ( H ) = − ( 3 2 [1 + (1 − r ) γ ] H + 1 α ) E, F$_{0}$ ( H ) = − 9 4 ( γ − 2) H 3 + 3 2 γ α H 2 E $^{2}$.