Euler and picard methods for solving first-order differential equations – classical and software-based

This paper describes and demonstrates the algorithms of Euler's and Picard's methods for solving first-order differential equations using a specific example. Both methods provide approximate solutions, with Euler's method [1], [3] yielding a numerical solution while Picard's method [2] initially provides an analytical solution, which can then be transformed into a numerical solution through direct substitution if necessary. The selected example illustrates these algorithms both through the classical solution approach and by utilizing the SageMath software [4]. The primary objective of this study is to highlight the advantages of numerical methods, particularly when modern technologies are employed.