Finally, we now able to derive an approximate estimation for g ( z ). According to Eq. (53) and (54), we get Amazingly, this expression looks the same as the one we obtained from the numerical ap- proximation with To solve Eq. (20) we divide the whole equation by y 3 and introduce a new variable v = 1 /y . Then Eq. (20) reads We then introduce the function v = w/x . Previous differential equation can be solved by separation of variables where C is an arbitrary constant of integration. To calculate the integral, we write the function to be integrated as Let us assume that Δ = 1 + 4 a > 0 (this implies that a > 0). Therefore the general solution of Eq. (58) can be written as leading to 19 g ( z ) ≈ (1 − r )( γ − 2) [1 + (1 − r ) γ ] 2 z (55) C = (1 − r )( γ − 2) [1 + (1 − r ) γ ] $_{2}$. (56) dv dx + v 2 + axv 3 = 0 . (57) x dw dx = w − w 2 − aw $^{3}$, (58) ∫ dw w − w 2 − aw 3 = ln C − $^{1}$x, (59) 1 w − w 2 − aw 3 = 1 w − aw aw 2 + w − 1 − 1 aw 2 + w − 1 . (60) ∫ dw w − w 2 − aw 3 = − 1 $_{2}$√$_{Δ}$ ln 2 aw − √$_{Δ + 1}$ 2 aw $_{+}$√$_{Δ + 1}$ − 1 2 ln ( aw 2 + w − 1 ) + ln w. (61) x = C w √$_{aw}$ 2 + w − 1 ( 2 aw $_{+}$√$_{Δ + 1}$ 2 aw − √$_{Δ + 1}$ ) 1 / $_{2}$√$_{Δ}$ , (62) y = 1 v = x w = C 1 √$_{aw}$ 2 + w − 1 ( 2 aw $_{+}$√$_{Δ + 1}$ 2 aw − √$_{Δ + 1}$ ) 1 / $_{2}$√$_{Δ}$ . (63) Appendix C: Solution of Abel equation y ˙ y − y = ax