  We study a singular nonlinear ordinary differential equation on intervals
$[0,R)$ with $R\le +\infty$, motivated by the Ginzburg-Landau models in
superconductivity and Landau-de Gennes models in liquid crystals. We prove
existence and uniqueness of positive solutions under general assumptions on the
nonlinearity. Further uniqueness results for sign-changing solutions are
obtained for a physically relevant class of nonlinearities. Moreover, we prove
a number of fine qualitative properties of the solution that are important for
the study of energetic stability.
