Chapter 2: Second-Order Linear Equations

Newton's second law is a second-order differential equation, and that single fact makes this chapter the physics chapter. Position determines force, force determines acceleration, and acceleration is the second derivative of position — so any mechanical system, from a car suspension to a bridge deck to an atom in a crystal, arrives as an equation relating $x^{″}$, $x^{\prime }$, and $x$. The same equation, with different names on the letters, governs the current in a circuit with a resistor, an inductor, and a capacitor.

The model problem is a mass on a spring: a restoring force pulling it back toward equilibrium, a damping force resisting its motion, and possibly an external force pushing on it from outside. Written out, that is $m{x}^{″}+c{x}^{\prime }+kx=F(t)$, and every behavior in the chapter comes from the interplay of those four ingredients. Two derivatives means two constants in the general solution, and therefore two initial conditions — where the mass starts and how fast it is moving.

Three phenomena carry the chapter, and each has a figure. First, the damping regimes: as resistance grows, oscillation that decays gives way to a return with no oscillation at all, and the crossover between them is sharp and nameable. Second, beats and resonance: pushing a system at nearly its own natural frequency produces a slow pulsing that grows without bound as the two frequencies converge — the reason soldiers break step on bridges. Third, the phase plane: plotting velocity against position instead of against time turns each solution into a single curve whose shape — closed loop or inward spiral — encodes the entire long-run behavior at a glance. That last idea is the bridge to Chapter 4.