Necessary conditions and a test for the stability of a system of two linear ordinary differential equations of the first order

We use the Riccati equation method for the derivation of necessary conditions and a test for the stability of a system of two linear first-order ordinary differential equations. We consider an example in which our results are compared with the results obtained by the Lyapunov and Bogdanov methods by estimating the norms of solutions via Lozinskii logarithmic norms, and by the freezing method.