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This PDF is an introduction to oscillatory systems through differential equations. It shows how starting from the simple harmonic oscillator, small mathematical tweaks (such as adding damping, nonlinear terms, or external forces) lead to a wide variety of new behaviors, including anharmonic motion, pendulum dynamics, and forced oscillations. You will learn how the structure of a differential equation determines the nature of motion, how solutions change under different conditions, and how to interpret these systems both analytically and visually through interactive phase portraits.

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