Solution of Third-Order Ordinary Linear Differential Equations in Discriminant Domains Implementing the Variational Iteration Method

Particularly in challenging discriminant cases, this study offers a fresh perspective on solving ordinary linear differential equations of third order. The method leverages the Variational Iteration Method by approximating the solution based on initial conditions and incorporating a correction functional with an optimized generic Lagrange multiplier using variational theory. The result is a reduction in computational complexity and the provision of an effective solution for these equations. The findings reveal that the proposed technique can yield approximate solutions with a minimal number of iterations, sometimes even with a single iteration, depending on the specific problem and initial conditions. This contribution significantly advances the field of differential equations, offering a fresh and efficient approach to tackle complex scenarios. We anticipate widespread adoption of this method for solving critical problems and for enhancing our comprehension of this fundamental area of mathematics.