Distributions of zeros of solutions for third-order differential equations with variable coefficients

For the third-order linear differential equations of the form r (t) x ′ ′ (t) ′ + p (t) x ′ (t) + q (t) x (t) = 0, we will establish lower bounds for the distance between zeros of a solution and/or its derivatives. The main results will be proved by making use of Hardy's inequality and some generalizations of Opial and Wirtinger type inequalities.