Lyapunov-type inequalities for a relativistic second-order differential equation

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 ẍ(t)+q(t)x(t)=0 where q(t) is a continuous function for t∈[a,b]. In the present note, a similar problem is solved for a differential equation of the form ddtẋ1−ẋ2+p(t)ẋ+q(t)x=0. The obtained inequality is applied to the estimate of the period of a periodic solution of relativistic differential Van der Pol equation.