CAUCHY PROBLEM FOR AN ORDINARY DIFFERENTIAL EQUATION: THEORETICAL FOUNDATIONS AND ANALYTICAL TECHNIQUES
Keywords:
ordinary differential equations, Lipschitz condition,Abstract
The Cauchy problem for ordinary differential equations (ODEs) plays a pivotal role in the theory and applications of differential equations. It involves solving a differential equation given an initial condition, often modeling time-dependent physical, biological, or economic systems. In this study, we rigorously examine the foundational aspects of the Cauchy problem, focusing on conditions for existence and uniqueness of solutions. Through theoretical results such as the Picard–Lindelöf theorem and Peano's existence theorem, we explore different solution behaviors. Furthermore, illustrative examples and computational approaches are provided to support the theoretical exposition. This work lays a solid groundwork for further studies in the theory of dynamical systems and numerical methods.Downloads
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