Skip to content
Free public beta until December 4, 2026: every course is open, no card needed. After that, the first chapter of each course stays free and full access is a one-time $39 ($19 for up to 100 founding members chosen for helpful feedback). Your progress is kept. How the beta works

differential-equations

Repeated real roots

taught

Taught in

Requires

Required by

Local knowledge map

Local knowledge map

Critically damped motion: the fastest non-oscillating return Energy in a vibrating mass-spring system 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 Cauchy-Euler with repeated and complex roots Damped free vibration: overdamped, critically damped, underdamped Long-term behavior and solution shape, read from the roots Repeated and complex roots at higher order Repeated real roots Complex roots and Euler's formula Distinct real roots Oscillatory solutions, amplitude-phase form Setting up the mass-spring equation of motion The Cauchy-Euler equation The characteristic polynomial at higher order The characteristic equation The general solution of a linear homogeneous equation

Click to expand

Solid arrows are required prerequisites; dashed are recommended or helpful. Lower rows are more foundational. Click any box to jump to that concept.