Our purpose is the design of efficient methods for solving stiff and nonstiff initial value problems for second-order ordinary differential equations of the special form y''=f(y). We analyse approximate factorization methods for second-order stiff differential equations, we construct general linear methods with parallel stages for nonstiff equations with peridic solution, we propose both parallel and sequential methods tuned to the special form of the solution. A number of convergence and stability results are derived and the performances of the methods are illustrated by means of a few examples from the literature.