We describe the efficient implementation of an explicit method to solve systems of stiff differential equations either on a grid or within a spectral approach. This method is based on an ansatz that approximates the solution. This ansatz depends on stiffness parameters that are shown to be related to the eigenfrequencies of the system. The accuracy and the performance of the method are tested in three different cases. First, we treat a highly stiff single differential equation, where explicit schemes converge rather slowly. Then, we solve the stationary Schr\"odinger equation associated to the quantum reflection of an ultracold atom by a surface. Finally, we consider the interaction of atomic hydrogen with a strong low-frequency laser pulse whose duration is of the order of 25 fs. We focus on the calculation of the above-threshold ionization electron spectrum, a problem which, under such realistic physical conditions, is computationally very demanding.