Solve differential equation: $f'''(x)=f(x)f'(x)f''(x)$

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.

Solution 1:

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

$$ \frac{df}{\sqrt{2\left(C_0\sqrt{\frac{\pi}{2}}\phi\left(\frac{f}{\sqrt 2}\right)+C_1\right)}} = dx $$