A theorem on energy integrals for linear second-order ordinary differential equations with variable coefficients

We aim at demonstrating a novel theorem on the derivation of energy integrals for linear second-order ordinary differential equations with variable coefficients. Namely, in this context, we will present a possible and consistent method to overcome the traditional difficulty of deriving energy integrals for Lagrangian functions that explicitly exhibit the independent variable. Our theorem is such that it appropriately governs the arbitrariness of the variable coefficients in order to have energy integrals ensured. In view of the theoretical framework in which the theorem will be embedded, we will also demonstrate that it can be applied as a mathematical method to solve linear second-order ordinary differential equations with variable coefficients. These results are expected to have a generalized fundamental character.