Solution to the differential equation $\frac{f}{f^\prime}=\frac{f^\prime}{f^{\prime\prime}}$

Question

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!

Answer 1

$\rm\bf Start$: Multiply through by $f\,''/f$ and integrate with respect to $x$: $$\frac{f\,''}{f\,'}=\frac{f\,'}{f} \implies \ln (f\,')=\ln f+C.$$ Now exponentiate and solve another differential equation similarly...

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$\rm\bf Finish$:

$$f\,'=e^Cf=Af\implies f(x)=Be^{Ax}.$$

Answer 2

$$ \frac{f}{f'} = \frac{f'}{f''} \qquad \Longleftrightarrow \qquad \frac{f'}{f} = \frac{f''}{f'} \qquad \Longleftrightarrow \qquad \int \frac{f'}{f} = \int\frac{f''}{f'} + C \qquad \Longleftrightarrow \qquad \ln f = \ln f' + C $$

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 $A = e^C$.