<p>This article presents an alternative to the standard solution method of linear second-order ordinary differential equations (ODEs) using integrating factors. This approach provides a clear and direct path to establish the duality between solutions and integrating factors of these ODEs. It explicitly exposes that an ODE can be solved using integrating factors if it is solvable by a known analytical method. This method proposes a sufficient and direct way to determine the required integrating factors of an analytically solvable ODE without solving its necessary adjoint equation and write the respective general solution as its integrating factors, leading to a more straightforward solution than the standard method. For demonstration, the method is applied to a variety of ODEs comprising well-known and special ODEs.</p>