Galileo watched a chandelier swing and found that tiny pushes, delivered in time with a pendulum's own rhythm, add up to a huge swing. Drive any oscillator at its natural frequency and each push arrives exactly when it helps — the response grows.
When the guessed particular solution already solves the homogeneous equation, undetermined coefficients fails and the fix is to multiply the guess by t. That extra t is not a trick — it is this runaway made visible: an oscillation whose height climbs linearly with time. The algebra and the growing swing are two faces of one fact.
Real structures snap before the math's infinity arrives. Tacoma Narrows (1940) twisted apart in a 40 mph wind; the modern account is aeroelastic flutter, not textbook resonance, but the lesson survives — an undamped system fed energy at its own beat has no ceiling.
Add even a little damping and the infinite spike becomes a tall but finite peak of height about 1/(2ζ). Detune a few percent and the runaway softens into beats — a slow breathing at the difference of the two frequencies.
Something in the simulation stopped unexpectedly — the lesson continues without it. You can move on; nothing you did was wrong.