I'm interesting in finding analytical solutions for the equation $$\alpha K x^\alpha + x -N = 0,$$ where $\alpha$ is a positive integer and both $K$ and $N$ are positive real constants.
Based on the meaning of the equation (derived from a problem in chemistry), there must be one (and only one) real solution between zero and $N$. Actually, I'm interested in only such solution, not any other outside the $(0, \, N)$ range.
I know that I can use numerical methods to solve for specific $\alpha$, $N$ and $K$. I'm seeking a more elegant method.
I also know that no general solution for a general polynomial of order 5 or more can exist, but maybe this one falls in some special case I'm not aware of.
I'm a chemist, so I apology if the question is malformed. Let me know if you need more details.