Solution to the iterative differential equation $-\gamma g^{\mathrm{\prime }}=g^{-1}$

Using a Picard-like operator $T$, we prove that the iterative differential equation $-\gamma g^{\mathrm{\prime }}=g^{-1}$ with parameter $\gamma >0$ has a solution $g=h\text{\hspace{0.17em}⁣}:[0,1]\to [0,1]$ for only one value $\gamma =\kappa \approx 0.278877$, and that this solution $h$ is unique. As an even stronger result, we exhibit $h$ as the global limit of the operator $T$.