Solution of differential equation $f\text{'}\left(x\right)f\text{'}\text{'}\text{'}\left(x\right)=3{(f\text{'}\text{'}\left(x\right))}^2$ is (where $y=f\left(x\right)$) 

 $\frac{3f\text{'}\text{'}\left(x\right)}{f\text{'}\left(x\right)}=\frac{f\text{'}\text{'}\text{'}\left(x\right)}{f\text{'}\text{'}\left(x\right)}$
 $\Rightarrow 3\ln f\text{'}\left(x\right)=\ln f\text{'}\text{'}\left(x\right)-\mathcal{l}nc$
 $\Rightarrow f\text{'}\text{'}\left(x\right)=c{(f\text{'}\left(x\right))}^3$
 $\frac{f\text{'}\text{'}\left(x\right)}{{(f\text{'}\left(x\right))}^3}=c$
we know that
 $\frac{d^{2}y}{d{x}^2}=-{\left(\frac{dy}{dx}\right)}^3.\frac{d^{2}x}{d{y}^2}$
 $\Rightarrow f\text{'}\text{'}\left(x\right)=-{(f\text{'}\left(x\right))}^3\frac{d^{2}x}{d{y}^2}$
 $\Rightarrow \frac{d^{2}x}{d{y}^2}=-c\Rightarrow \frac{dx}{dy}=-c_{1}y+c_2$
 $x=\frac{-c_1}{2}{y}^2+c_{2}y+c_3$