On Some Analytical Methods for Solving Second Order Ordinary Differential Equations with Constant Coefficients

Second- order differential equations have great significance theoretically and practically. The article begins by providing an in-depth analysis of different methods for solving homogeneous second-order ordinary differential equation(ODEs) involving constant coefficients, the concepts of solution and the different methods of obtaining these solutions are analyzed. This study reviews the classical techniques, including the characteristic equation methods, solution involving real roots, repeated roots and complex roots and presents the general solution for each case, with examples. Next, the article goes to discuss use of the method undetermined coefficients for solving non-homogeneous equations. The article also highlights the role of initial conditions in determining particular solutions. By providing a systematic overview of these approaches the researcher contributes to a deeper understanding of the structure and application of the second -order linear ODEs, offering both theoretical insights and practical problem-solving tools.