The Riccati equation method is used to establish an oscillatory and a non oscillatory criteria for nonhomogeneous linear systems of two first-order ordinary differential equations. It is shown that the obtained oscillatory criterion is a generalization of J. S. W. Wong's oscillatory criterion.

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The method of Riccati equations is used to study some properties of third order linear ordinary differential equations. Some criteria of asymptotic behavior and non stability of solution of this equation are obtained. Two oscillatory criteria are proved. 2010 AMS Mathematics subject classification. Primary 34C10, 34C99.

A new approach with the Riccati equation method is used to obtain a non oscillation criterion for extended quasi linear Hamiltonian systems.

Some global existence criteria for quaternionic Riccati equations are established. Two of them are used to prove a completely non conjugation theorem for solutions of linear systems of ordinary differential equations.