Behavior of stabilizing solutions of the Riccati equation

Sufficient conditions are found for the existence of stabilizing solutions of the Riccati differential equation $y'=\bigl(y-y_1(x)\bigr)\bigl(y-y_2(x)\bigr)$ with given $y_1(x)$ and $y_2(x)$. For various types of stabilizing solutions, the number of points of extremum is examined.