How to solve a given differential equation of form $\frac{d}{dx}\left(f(x)^n\right)$? We have a differential equation of the form $$\frac{d}{dx}\left(f(x)^n\right)=f(nx).$$ By inspection we have $f(x)=\sin(x)$ for $n=2$. Are there any other solution?

To elaborate on my comment:  Let's look for a solution of the form $f(x) = C e^{\alpha x}$ for some (possibly complex) constants $\alpha$ and $C$.  Then we have
\begin{align*}
\frac{d}{dx} \left[ \left( C e^{\alpha x} \right)^n \right] &= C e^{\alpha (nx)} \\
\frac{d}{dx} \left[ C^n e^{\alpha n x} \right] &= C e^{\alpha nx} \\
C^n \alpha n e^{\alpha n x} &= C e^{\alpha n x} 
\end{align*}
which is satisfied if $C^{n-1} \alpha n = 1$, i.e., $f(x) = C e^{C^{1-n} x/n}$ for any constant $C$.  
