Differential Equations and Applications

This course provides a comprehensive study of ordinary differential equations (ODEs), focusing on both theoretical concepts and solution techniques. It begins with the classification and solution of linear and non-linear first-order ODEs, followed by an in-depth exploration of second-order ODEs, including the concepts of linear dependence and independence of solutions. Various methods for solving second-order equations are introduced, such as the method of undetermined coefficients, variation of parameters, and reduction of order. The course also covers advanced techniques like the Laplace Transform for solving initial value problems and the application of Fourier series for representing periodic solutions. Sturm-Liouville problems are studied as a framework for solving boundary value problems and understanding eigenfunction expansions. Additionally, the course addresses the solution of systems of linear ODEs using matrix methods and eigenvalue analysis. This course equips students with the mathematical tools necessary to model and analyze real-world dynamic systems in science and engineering.