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differential-equations

Complex roots and Euler's formula

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Taught in

Requires

Required by

Local knowledge map

Local knowledge map

Classifying centers Critically damped motion: the fastest non-oscillating return Energy in a vibrating mass-spring system Long-term behavior and solution shape, read from the roots Overdamped motion: two real decaying exponentials Steady-state versus transient response Superposition and the general solution at $n$th order The damping ratio The mechanical-electrical analogy Undamped free vibration and natural frequency Cauchy-Euler with repeated and complex roots Complex Fourier series Damped free vibration: overdamped, critically damped, underdamped Oscillatory solutions, amplitude-phase form Repeated and complex roots at higher order The eigenvalue method -- complex eigenvalues Complex roots and Euler's formula Distinct real roots Repeated real roots Setting up the mass-spring equation of motion The Cauchy-Euler equation The characteristic polynomial at higher order The eigenvalue method -- distinct real eigenvalues The full Fourier series The characteristic equation The general solution of a linear homogeneous equation

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