W-transform for exponential stability of second order delay differential equations without damping terms

Abstract In this paper a method for studying stability of the equation x ″ ( t ) + ∑ i = 1 m a i ( t ) x ( t − τ i ( t ) ) = 0 $x^{\prime \prime }(t)+\sum_{i=1}^{m}a_{i}(t)x(t- \tau_{i}(t))=0$ not including explicitly the first derivative is proposed. We demonstrate that although the corresponding ordinary differential equation x ″ ( t ) + ∑ i = 1 m a i ( t ) x ( t ) = 0 $x^{\prime \prime}(t)+\sum_{i=1}^{m}a_{i}(t)x(t)=0$ is not exponentially stable, the delay equation can be exponentially stable