Solving $x^5-x-a=0$ by inverse function theorem

Question

If you can find a differential equation of the form

$$y'=F(y)$$ for a quintic $y$, its inverse would be expressed as

$$x=\text{const.}+\int\frac{dy}{F(y)}.$$

Differentiating any quintic yields a quartic, and by eliminating between the quintic and its derivative, we may have it such that the coefficients work out nicely in order to isolate such an explicit function as $F(y)$ from some $\Phi(y,y')=0$. For example, in the quadratic case

$$y=x^2+ax+b$$ we have

$$y'=2x+a\iff x=\frac{y'-a}{2}$$ and therefore,

$$4y=y'^2+4b-a^2$$ so

$$x=\int \frac{dy}{\sqrt{4y+a^2-4b}}$$ very conveniently. Thus my question: can the coefficients be manipulated for this to work. Are there enough "degrees of freedom" so to speak?

Answer 1

To answer my own question: No, in general $$y'=F(y)$$ cannot be recovered practically from $$\Phi(y,y')=0.$$ In fact, in some sense, each requires solving the original polynomial equation anyway.