Hartman and Lyapunov Inequalities

This undergraduate thesis addresses the Hartman and Lyapunov inequalities for second-order linear differential equations. First, the Green's function method for solving linear differential equations satisfying Dirichlet boundary conditions is comprehensively explained, and the integral forms of the solutions are obtained through this method. This approach facilitates both a clearer understanding of theoretical approaches and plays a significant role in the proof of inequalities. Hartman's inequality provides a lower bound on the definite integral of the potential function, depending on the behavior of the solution between zeros. Lyapunov's inequality, on the other hand, provides a tighter bound for the same problem, providing important information about the stability and behavior of the solution. The relationship between these two inequalities is analyzed in detail. A numerical example is also provided to demonstrate the validity of both Hartman and Lyapunov inequalities. Calculations based on this example demonstrate the accuracy of the theoretical results. The findings demonstrate how such inequalities can be used in the analysis of boundary value problems in the theory of differential equations.