Analytical solution for nonlinear equation

Question

Simple question:

Does $\alpha = \frac{x}{\beta} - \left(\frac{x}{\gamma}\right) ^{1/\delta}$ have an analytical solution? ($\alpha,\beta,\gamma,\delta$ are constant)

I'm working on big data arrays and either need to solve this equation analytically or spend resources crunching away at least squares, iterations etc.

It comes from the Sersic light profile of galaxies which is talked about here (Equation 6). I need to find where the difference between galaxy components is less than a certain value

So, I'm trying to solve: $\mu_{bulge} - \mu_{exp-disc} - \rm{limit} = 0$ ($\mu$s are from equation 6)

By looking at the profiles, it's obvious that this equation has zero to two real solutions (bulge overlapping the disc

Answer 1

For certain values of the constants, yes. In general, no.

Answer 2

If $\delta$ is a rational number, $x$ is linked with the constants by algebraic operations. The equation is related to a polynomial equation of $x$ then.