Weak-Coppel problem for a class of Riccati differential equations

We study the $T$-periodic solutions of the real Riccati differential equation $x^{\mathrm{\prime }}=x^2+\gamma (t),$ where $x=x(t)$ and $\gamma$ is a $T$-periodic function. Our goal is to define a real-valued discriminant ${\mathrm{\Delta }}_{\gamma }$ that determines whether the equation admits two, one, or no $T$-periodic solutions, in analogy with the classical discriminant of the quadratic algebraic equations. This problem is closely related to a question posed by Coppel concerning the characterization of bifurcation curves for planar quadratic differential systems. Although our result does not cover all periodic Riccati differential equations, many of them can be transformed into this particular form.