Advanced Analysis II: Cauchy Problem Resolution

This lecture focuses on the numerical resolution of a Cauchy problem using the method of separation of variables. The instructor presents a typical differential equation, where the derivative of U with respect to t is equal to t² times (1 + U²). The lecture explains how to manipulate the equation by isolating U and integrating both sides. The instructor emphasizes the importance of understanding the theoretical construction of the solution before applying it practically. The integration process is detailed, showing how to derive the general solution and the significance of the initial condition in determining the constant. The lecture concludes with a discussion on the interval of definition for the solution, highlighting that the solution is local rather than global, depending on the behavior of the function involved. The instructor provides insights into ensuring the solution remains well-defined within a specific interval, ultimately leading to a comprehensive understanding of the Cauchy problem resolution.