I believe there is no closed form expression but with a bit of pen-and-paper maths you can get this into a form where you can easily compute very good numeric approximations. I only looked at the first equation, the others seems in general simiar.

Start with the substitution $s=\sqrt{2q^2/Mr}$ then $r=\frac{2q^2}{Ms^2}$. Plug this into the equation and simplify yields

$$\frac{q^2}{4M^2} = s^2\frac{e^{-s}}{1+e^{-s}}$$

Now note that the right side of the equation does not contain any of the parameters so for any given values of $q$ and $M$ you can compute all possible solutions for $s$ numerically to high accuracy.

I believe there is no closed form expression but with a bit of pen-and-paper maths you can get this into a form where you can easily compute very good numeric approximations. I only looked at the first equation, the others seems in general simiar.

Start with the substitution $s=\sqrt{2q^2/Mr}$ then $r=\frac{2q^2}{Ms^2}$. Plug this into the equation and simplify yields

$$\frac{q^2}{4M^2} = s^2\frac{e^{-s}}{(1+e^{-s})^2}$$

Now note that the right side of the equation does not contain any of the parameters so for any given values of $q$ and $M$ you can compute all possible solutions for $s$ numerically to high accuracy.