<p>The feasibility of using a problem-dependent method to solve systems of second order ODEs is corroborated by an eigen-based theory and a methodology to develop such a numerical method is constructed. The key steps of this methodology are to decouple a system of ODEs of second order into a set of uncoupled ODEs of second order; next, an eigen-dependent method is proposed to approximate the solution of each uncoupled ODE of second order. It is vital to transform all eigen-dependent methods to a problem-dependent method to bypass an Eigen analysis. The development of an eigen-dependent method plays a key role in this methodology so that slow eigenmodes can be accurately integrated while there is no instability or excessive amplitude growth in fast eigenmodes. This can explain why a problem-dependent method can simultaneously combine the explicitness of each step and A-stability. Consequently, huge computational efforts can be saved for solving nonlinear stiff problems. A new family of problem-dependent methods is developed in this work so that the feasibility of the proposed methodology can be affirmed. It has almost the same performance as that of the HHT-<i>α</i> method. However, it can save more than 99.5% of CPU demand in approximating a solution for a system of 1000 nonlinear second order ODEs.</p>