In some situations, especially if one demands the solution of the differential equation with a great precision, it is preferable to use high-order methods. The methods considered here are similar to Runge—Kutta methods, but for the second-order equation y″= f(x, y). As for Runge—Kutta methods, the complexity of the order conditions grows rapidly with the order, so that we have to solve a non—linear system of 440 algebraic equations to obtain a tenth—order method. We demonstrate how this system can be solved. Finally we give the coefficients (20 decimals) of two methods with small local truncation errors.