This technical note is concerned with exponential stability analysis of linear time-varying systems represented by the second-order vector differential equations with the singular leading coefficient matrix. Using bounding techniques on the trajectories of the linear time-varying systems, the stability problem of the time-varying systems is transformed to that of the time-invariant systems and a new sufficient condition for exponential stability is derived. The obtained condition can be applied to the nonsingular case and it is proven that the condition for the nonsingular case is superior to a test presented in the recent literature. Analogously, the proposed method is also extended to the stability analysis for a class of linear time-varying systems with delay. By illustrative examples, better results are obtained by the proposed criteria as compared with some results in the literature.