Abstract <p>Some linear second-order differential algebraic equations (DAEs) are considered on a finite interval of integration with given initial data. A class of problems with a unique sufficiently smooth solution is formulated in terms of matrix polynomials. We assume that solutions to the problems may contain stiff and rapidly oscillating components. The main challenges in developing algorithms for numerical solution of this class of problems are highlighted. We propose to represent the original problem in the form of a system of integral differential or integral equations with an identically singular matrix as a coefficient of its main term for constructing effective methods of numerical solution of the second-order DAEs. We construct some numerical solution methods for the problems represented in this way. These algorithms are based on explicit Adams quadrature formulas for calculating the integral term and extrapolation formulas for the other terms. The results of test calculations are presented and analyzed.</p>