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!
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!
$\rm\bf Start$: 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...
$\rm\bf Finish$:
f\,'=e^Cf=Af\implies f(x)=Be^{Ax}.
\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 $A = e^C$.