From First to Second Order

Differential Equations · Axiom Academy

One extra derivative changes everything. First-order equations only know how to decay — second-order equations can swing back and forth forever. Watch that one difference play out. Same starting point, opposite fates Both curves below start at the exact same place: y(0)=1 . One solves a first-order equation, y' + y = 0 . The other solves a second-order equation, y'' + y = 0 . Same starting height — but watch what happens as time moves forward. Press play. The first-order solution y=e^ -t decays straight to zero and stays there. The second-order solution swings back past zero, up to -1 , and keeps oscillating — forever. By , the decay curve has vanished to nearly zero — the oscillation is back up at its peak, exactly as strong as when it started. The coefficient controls the oscillation's speed This is a pure second-order equation, y'' + ky = 0 , with solution — think of a spring or a pendulum with no friction at all. Drag k and watch the curve oscillate faster or slower. It never decays, no matter what you pick. The period is — a stiffer spring (bigger k ) oscillates faster, with a shorter period. But the swing never dies out. One extra term bridges the two worlds Add a damping term: y'' + cy' + y = 0 . Drag c and watch the SAME equation cross through three regimes — light damping still oscillates as it decays, heavy damping decays with no oscillation at all, matching the pure first-order behavior you saw at the very top.

This is the written version of the interactive lesson above. See the full Differential Equations course.