Consider this polynomial equation:
$$(n+1)~x^{2n+1}-n~x^{2n}-n=0,~~~~n \geq 2,~~~n \in \mathbb{N}$$
It's related to another question of mine, but I don't think the context matters here.
I'm interested in the positive real solution. For $n=1$ the positive real solution is $x=1$.
What I want to ask:
Is it possible for the above equation to be solved in radicals for some $n \geq 2$ and what is the explicit expression for the real positive solution?
If we substitute $x=\frac{1}{y}$, we obtain the equation:
$$y^{2n+1}+y-\frac{n+1}{n}=0$$
