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How should our students think about external forcing in differential equations setting, and how can we help them gain intuition? To address this question, we share a variety of problems and projects that explore the dynamics of the undamped forced spring–mass system. We provide a sequence of discovery-based exercises that foster physical and mathematical intuition about polynomial forcing, as we build tools and techniques (including Green's function) to explore the amazing behavior of We encourage the insightful use of a Computer Algebra System, and provide a paired supplemental Maple worksheet. [ABSTRACT FROM AUTHOR]

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 equationand. 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. [ABSTRACT FROM PUBLISHER]

This article investigates the relationship between an unperturbed integrodifferential system and a perturbed integrodifferential system that have different initial positions and an initial time difference (ITD). Variation of parameter techniques are employed to obtain integral formulae and to define Lyapunov-like functions. [ABSTRACT FROM AUTHOR]

We consider a Volterra discrete system with nonlinear perturbation x ( n +1)= A ( n ) x ( n )+ ∑ s =0 n B ( n , s ) x ( s )+ g ( n , x ( n ) and obtain necessary and sufficient conditions for stability properties of the zero solution employing the resolvent equation coupled with the variation of parameters formula. [ABSTRACT FROM AUTHOR]