On the Lyapunov type inequality

A.M. Lyapunov proved the inequality that makes it possible to estimate the distance between two consecutive zeros $ a $ and $ b $ of solutions of a linear differential equation of the second order $ x''(t) + q (t) x (t) = 0$ where $ q (t) $ is a continuous function for $ t \in [a, b] $. In the present note, a similar problem is solved for a linear differential equation of the form $ x'' (t) + p (t) x'(t) + q (t) x (t) = 0 $. The obtained inequality is applied to the periods estimate of periodic solutions of nonlinear differential Liénard and Van der Pol equations.