On the stability of systems of two first-order linear ordinary differential equations

We use the Riccati equation method to establish necessary and sufficient stability conditions and a stability criterion for a system of two first-order linear ordinary differential equations. We present examples in which these results are compared with the results obtained by the Lyapunov and Bogdanov methods, by a method involving estimates of solutions in the Lozinskii logarithmic norms, and by the freezing method.