We differentiate both sides of $(\star )$ and obtain

${\displaystyle f(x)=3f\text{'}(x).}$

Thus if there is a solution of $(\star )$ it must be of the form

${\displaystyle f(x)={\mathrm{Ae}}^{x/3},}$

for some constant $A$. We check to see whether or not this is a solution. Thus with $f(x)={\mathrm{Ae}}^{x/3}$ we have

${\displaystyle {\int }_0^x{\mathrm{Ae}}^{t/3}\hspace{0.5em}\mathrm{dt}=[3{\mathrm{Ae}}^{t/3}{]}_0^x=3{\mathrm{Ae}}^{x/3}-3A.}$

Thus $f(x)={\mathrm{Ae}}^{x/3}$ is a solution when (and only when) $A=-k/3$. Thus the unique solution is

${\displaystyle f(x)=\frac{-k}{3}{e}^{x/3}.}$