Laplace Transform: Solving Differential Equations

This lecture focuses on the application of the Laplace transform to solve differential equations. The instructor begins by summarizing previous concepts and introduces the Laplace transform as a method for handling differential equations, particularly those defined for positive time. The lecture covers the properties of the Laplace transform, including its linearity and the handling of initial conditions. Several examples are provided, including the resolution of a first-order differential equation and a second-order system involving mass-spring-damper dynamics. The instructor demonstrates how to apply the Laplace transform to these equations, deriving solutions and discussing the uniqueness of solutions. The lecture also touches on the convolution theorem and its implications for solving differential equations. Throughout, the instructor emphasizes the importance of understanding the conditions under which the Laplace transform is applicable and the significance of initial conditions in determining unique solutions. The session concludes with a discussion on the inversion of the Laplace transform and its relevance in practical applications.