TITLE: Solution to the differential equation $\frac{f}{f^\prime}=\frac{f^\prime}{f^{\prime\prime}}$
QUESTION [5 upvotes]: I am wondering if anyone could please post the solution to the following differential equation for the function $f(x)$:
$$\frac{f}{f^\prime}=\frac{f^\prime}{f^{\prime\prime}}$$
Thanks!

REPLY [6 votes]: $$
\frac{f}{f'} = \frac{f'}{f''} \qquad \Longleftrightarrow \qquad \frac{f'}{f} = \frac{f''}{f'} \qquad \Longleftrightarrow \qquad \int \frac{f'}{f} = \int\frac{f''}{f'} + C \qquad \Longleftrightarrow  \qquad \ln f = \ln f' + C
$$
Hence, taking exponentials on both sides,
$$
f = K f' \ ,
$$
where $K = e^C$. Renaming $K$ as $\frac{1}{K}$, this is the same as
$$
\frac{f'}{f} = K \qquad \Longleftrightarrow \qquad \int \frac{f'}{f} = \int K + C  \qquad \Longleftrightarrow \qquad \ln f = Kx + C \qquad \Longleftrightarrow \qquad f(x) = A e^{Kx}
$$
where $A = e^C$.