This paper deals with a new version of the Picard-Lindelöf method for solving the Cauchy problem dudt=φ(u(t),t),u(t0)=u0, where the solution u(t) must be found in the interval [t0,tK]. The approach discussed in this study is based on two-sided a posteriori estimates derived by A. M. Ostrowski (1972). Furthermore, it is stated that the APL (Adaptive Picard-Lindelöf) algorithm approximates the solution to within any given tolerance.