This research focuses on the application of the Finite Element Method (FEM) for obtaining numerical solutions to linear differential equations that arise in various scientific and engineering problems. The method works by discretizing the problem domain into smaller subdomains called elements, which are connected at nodes. Unknown variables, such as displacement or temperature, are approximated at these nodes using suitable shape functions. A weak formulation of the governing equations is developed, and element-level equations are assembled into a global system. Appropriate boundary conditions are incorporated to represent real physical situations. The resulting system is then solved using numerical techniques to obtain approximate solutions. FEM is widely used in structural analysis, mechanical design, heat transfer, fluid flow, and biomedical engineering The results obtained from FEM provide reliable approximations that support better design, improve safety, and reduce dependence on experimental testing. This study highlights the effectiveness and versatility of FEM as a powerful computational tool for solving real-world engineering problems involving differential equations.