Variation of Parameters in Differential Equations (A Variation in Making Sense of Variation of Parameters)

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The method of variation of parameters can be found in most undergraduate textbooks on differential equations. The method leads to solutions of the non-homogeneous equation of the form , a sum of function products using solutions to the homogeneous equation and . Why though, might varying parameters of solutions to the homogeneous equation be a feasible approach? An insight is needed, even for a good guess. This paper is for instructors teaching a first course in undergraduate differential questions, where the students already have covered first-order linear equations and second-order homogeneous linear equations. We provide a variation to the standard textbook approach to variation of parameters, intended to help the student find their way toward a key insight: “Aha! Let's try …” We point to several reasons why it can be important for students to learn variation of parameters.