Auxiliary equations with complex roots, for 2nd order linear differential equations

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From a channel Lumen includes in full: Steve Chow (blackpenredpen) is a long-time calculus teacher who works integrals, derivatives, series, special functions and precalculus problems in full on the whiteboard, including multi-hour practice marathons.

A 11-minute video from blackpenredpen's catalogue; the title names its topic.

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Part of Auxiliary equations with complex roots, (Nagle Sect4.3)