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## A. Solution in the negative region ( x < 0). 

Using the variable y = − e − a ( x + L ) and the transformation φ = y $^{µ}$(1 − y ) − $^{λ}$f ( y ) used in [24] we will show that it is possible to obtain an exact solution in the form of an hypergeometric function by imposing appropriate constraints on the mass function. With the above transformation and using Eq. (8), Eq. (7) becomes: 

<img src='content_image/367690.jpg'>

Here and in the following the dot indicates derivation with respect to the transformed variable ( ˙ m = dm/dy ). In order to keep the structure of the hypergeometric differential equation we impose the following condition on this term: 

which has the following solution ( m$_{0}$ integration constant): 

With this choice of mass function Eq. 10 becomes: 

<img src='content_image/367691.jpg'>

<img src='content_image/367692.jpg'>

<img src='content_image/367693.jpg'>

In order that Eq. 13 keeps the structure of the hypergeometric differential equation as in the m = const case we may impose the following conditions on the mass function: 

<img src='content_image/367694.jpg'>

