I'd love your help with finding the function $y$ which satisfies: y'=y^a, $y(a)=a-2$, for $a \in \mathbb{N}$
This is what I did:
\begin{align*} \int \frac{y'}{y^a}dx&=\int 1dx\\ \frac{y^{-a+1}}{-a+1}+c_1&=x+c_2\\ y^{-a+1}&=(x+C)(-a+1), \end{align*} where $C=c_2-c_1$, and for $a=1$ there's no solution.
So I get $y=\left(\frac{1}{(x+c)(1-a)}\right)^{a-1},$ finding $c$ is not pleasant.
I assume that something is wrong, Am I suppose leave $y$ in the way that I find it after finding $c$?
Thanks a lot!