I have the equation $\frac{\partial A}{\partial t}+cA^m\frac{\partial A}{\partial s}=0$. I need to rearrange so that the coefficient of the $\frac{\partial A}{\partial s}$ term is equal to 1.
Since the partial derivative term of interest is with respect to $s$, I figured that I could treat $cA^m$ as a constant with respect to $s$, leading to $\frac{\partial A}{\partial t}+\frac{\partial (cA^{m+1})}{\partial s}=0$. But the correct rearrangement is apparently $\frac{\partial A}{\partial t}+\frac{\partial \left(\frac{cA^{m+1}}{m+1}\right)}{\partial s}=0$. I am struggling to see why I have to divide $cA^{m+1}$ by $m+1$.
Why would you even think that you can treat $cA^{m+1}$ as a constant with respect to $s$? Think of it like in ODEs - $$\frac{\partial \left(\frac{cA^{m+1}}{m+1}\right)}{\partial s} = \frac{\partial \left(\frac{cA^{m+1}}{m+1}\right)}{\partial A}\frac{\partial A}{\partial s} = cA^m\frac{\partial A}{\partial s}$$