Linear ordinary differential equations

First- and second-order, homogeneous and forced.

Differential equations are how physics talks about change. Almost every quantity in this bookshelf — the pressure at a point in air, the displacement of a vibrating mass, the receptor potential of a sensory cell, the charge on a capacitor — evolves in time according to an equation that involves not just the quantity itself but its rate of change. Those are differential equations, and the simplest, most useful family is the linear ones with constant coefficients.

This chapter is built as a gentle ramp in four lessons. We start with the picture (an ODE is a field of slopes), then learn the single algebraic trick that solves the whole family, then watch that trick play out in the canonical examples — first decay, then oscillation, then damping, then forcing. The closing lesson zooms out to the geometric phase-plane view that ties everything together.

If you have not thought about ODEs in a while, that is the audience this chapter is written for. We will reintroduce every idea before using it.

• 5.1 What is an ODE? — slope fields, and what “order” means.
• 5.2 First-order linear ODEs — exponential decay, and the algebraic trick that solves every linear ODE.
• 5.3 Second-order linear ODEs — oscillation, damping, resonance.
• 5.4 Phase plane and classification — the geometric view, equilibrium classification, and reference notes.