How to solve a differential equation problem?

Question

I want to find the solution of the following differential equation $$\frac{dy}{dx}+y^m=1.$$ For example, if $m=1$, then $y=1-e^{-x}.$ If $m=2$, we have $y=\tanh(x)$, but for $m\ge 3$, $y=?$.

Answer 1

We have that $dx =\frac{1}{1-y^m}dy$, so $x=\int \frac{1}{1-y^m}dy$, which has no elementary representations. (Can still be expressed in terms of hypergemoetric or incomplete beta, but I don't think that was what you were looking for.)