Why is the solution to $y' = y^n$ always in polynomial form EXCEPT when $n = 1$?

Question

Could someone explain (intuition-wise) why the differential equation

$$y' = y^n$$

for $n \in \mathbb{N}$ seems to always some kind of polynomial solution (or a ratio of polynomials, etc.) except when $n = 1$, in which case the solution seems to be exponential?

What's so special about $n = 1$ that (if you'll pardon the term) differentiates it from e.g. $n = 2$?

Answer 1

If you rewrite equation: $$\frac{dy}{y^n}=dx,$$ and then $$\int_{y_0}^y \frac{dy}{y^n}=x-x_0,$$ you should take integral. But case $n=1$ is special — integral is $\ln y$, not a $\frac{y^{1-n}}{1-n}$.