THE FORMULATION OF THE CAUCHY PROBLEM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS AND ITS THEORETICAL FOUNDATIONS

This article examines the formulation and theoretical foundations of the Cauchy problem for first-order differential equations, which occupies a central role in mathematical analysis and its applications. The study begins with a rigorous definition of first-order differential equations and the corresponding initial value problem, emphasizing the conditions under which a unique solution exists. Particular attention is devoted to the fundamental existence and uniqueness theorems, including the classical results associated with Lipschitz continuity and continuity conditions. The paper further explores the geometric interpretation of solutions, highlighting the relationship between integral curves and direction fields. In addition, the analytical framework underlying the Picard–Lindelöf theorem is discussed in detail, providing insight into iterative methods for constructing solutions. The role of continuity, boundedness, and differentiability in ensuring well-posedness is also critically analyzed. Through a systematic exposition of these theoretical principles, the article establishes a coherent understanding of the Cauchy problem, demonstrating its significance in both theoretical investigations and practical problem-solving across various fields of science and engineering.