order of a differential equation
Volume I named the term 'differential equation' — an equation that relates an unknown function to its own derivatives — and then mostly walked away from it. The very first question to ask about any such equation is: how many times deep does the differentiation go? That count is the order. It is the single most important label on a differential equation, the way 'degree two' is the headline fact about a quadratic.
Precisely, the order is the highest derivative that appears. The equation dy/dx = x*y has order one, because dy/dx is as high as it goes. The equation d^2y/dx^2 + omega^2 y = 0 (the harmonic oscillator) has order two, because of the second derivative. An equation containing d^3y/dx^3 is third order even if a first derivative also appears — only the highest one counts. This whole field studies first-order equations: those expressible as dy/dx = f(x, y), one derivative deep.
Order matters because it predicts how much freedom the solution carries. A first-order equation needs one extra condition to pin down a unique solution (say, the value of y at one point); an n-th order equation needs n. That is why a first-order initial-value problem fixes y at a single instant, while a second-order one — a vibrating mass — needs both an initial position and an initial velocity. Order is the bookkeeping that tells you how many constants of integration you must chase.
Newton's law for a falling object, m*d^2x/dt^2 = -m*g, is second order (a second derivative, acceleration). Radioactive decay, dN/dt = -k*N, is first order. The order is read off by inspection, not by solving.
Order = the highest derivative present, read directly from the equation.
Order is about the highest derivative, never about the highest power. (x*y')^5 = y is first order (only y' appears), even though it is raised to the fifth power — that fifth power is the degree, a separate label.