Obrechkoff methods having additional parameters for general second-order differential equations

A class of two-step implicit methods involving higher-order derivatives of y for initial value problems of the form y″ = f(t, y, y′)is developed. The methods involve arbitrary parameters p and q, which are determined so that the methods become absolutely stable when applied to the test equation y″ + λy′ + μy = 0. Numerical results for Bessel's and general second-order differential equations are presented to illustrate that the methods are absolutely stable and are of order O(h⁴), O(h⁶) and O(h⁸).