The O. D. E. I am trying to solve is $y''(-y')^m=-y^ny',\quad y(1)=1,\quad y'(1)=-1,\quad (*)$ where $m$ and $n$ are integers greater than one. If we observe that $y''(-y')^m=-\frac{1}{m+1}[(-y)^{m+1}]',$ and that $y^ny'=\frac{1}{n+1}[y^{n+1}]',$ we can integrate the equation ($*$) and obtain the following equation: $-\frac{1}{m+1}(-y')^{m+1}+\frac{1}{n+1}(y)^{n+1}=C$ where $C$ is a constant. From now on, I don't know what to do.
Could someone give me an idea to complete the solution above or show me another way to find $y$ in the equation ($*$)?