content_image/354719.jpg content_image/354720.jpg content_image/354721.jpg content_image/354722.jpg content_image/354723.jpg content_image/354724.jpg content_image/354725.jpg content_image/354726.jpg content_image/354727.jpg
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 

## Appendix C: Solution of Abel equation y ˙ y − y = ax 

<img src='content_image/354719.jpg'>

<img src='content_image/354720.jpg'>

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 

<img src='content_image/354721.jpg'>

<img src='content_image/354722.jpg'>

<img src='content_image/354723.jpg'>

where C is an arbitrary constant of integration. To calculate the integral, we write the function to be integrated as 

<img src='content_image/354724.jpg'>

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 

<img src='content_image/354725.jpg'>

<img src='content_image/354726.jpg'>

<img src='content_image/354727.jpg'>

