TITLE: Solve differential equation: $f'''(x)=f(x)f'(x)f''(x)$
QUESTION [10 upvotes]: I came across $f'''(x)=f(x)f'(x)f''(x)$ but I don't know how to solve it.
I tried 
$\frac{f'''(x)}{f''(x)}=f(x)f'(x)$
$\ln|f''(x)|=\frac{1}{2}f(x)^2+c_{1}$
But from there I have no idea how to proceed.
Please help me solve this if possible.

REPLY [5 votes]: After
$$
f'' = C_0e^{\frac 12 f^2}
$$
we have
$$
f'' f' = C_0e^{\frac 12 f^2}f'\Rightarrow \frac 12(f')^2=C_0\sqrt{\frac{\pi}{2}}\phi\left(\frac{f}{\sqrt 2}\right)+C_1
$$
with
$$
\phi\left(x\right)=\int_0^x e^{\zeta^2}d\zeta
$$
and finally we arrive to the solution after integrating
$$
\frac{df}{\sqrt{2\left(C_0\sqrt{\frac{\pi}{2}}\phi\left(\frac{f}{\sqrt 2}\right)+C_1\right)}} = dx
$$