A class of Runge–Kutta methods directly applicable to second-order equations of the form y″ = f(x,y,y′), a ≤ x ≤ b, y(a) = y0, y′(a) = y0′ without requiring a reduction to first-order systems (see ordinary differential equations). Extrapolation methods and linear multistep methods of this direct type have also been developed. Such methods can be particularly advantageous for equations of the type y″ = f(x,y), where y′ does not appear explicitly.
y″ = f(x,y,y′), a ≤ x ≤ b,
y(a) = y0, y′(a) = y0′