A functionally-fitted Numerov-type method is developed for the numerical solution of second-order initial-value problems with oscillatory solutions. The basis functions are considered among trigonometric and hyperbolic ones. The characteristics of the method are studied, particularly, it is shown that it has a third order of convergence for the general second-order ordinary differential equation, y''=f left( x,y,y' right) , it is a fourth order convergent method for the special second-order ordinary differential equation, y''=f left( x,yright) . Comparison with other methods in the literature, even of higher order, shows the good performance of the proposed method.