We study mathematical perturbed equations that can be used to describe how learner motivation, engagement, and proficiency develop together in English as a Foreign Language (EFL) contexts. The basic system uses first-order ordinary differential equations to show how motivation and engagement influence each other, while proficiency grows in a logistic way but also declines if not reinforced. To try to make the system closer to classroom situations, we add a small perturbation term that represents changes caused by teaching methods or other external factors. We analyze the stability and behavior of the system. Numerical examples show that the equations remain stable even under small perturbations. Our results suggest that mathematical systems with perturbations provide useful tools for understanding how motivation and learning progress in second language acquisition.