Differential Equation

An equation which contains the derivative of a function is called a DIFFERENTIAL EQUATION. This definition is deliberately broad: it says nothing about which variable is independent, which is dependent, or how many derivatives (or of what order) must appear — only that at least one derivative must be present. The chapter immediately backs this up with six varied examples so the definition is not read too narrowly:

1. dy/dx = cos x — the simplest possible case, a first-order equation with y as a function of x.
2. d²y/dx² + k²y = 0 — a second-order equation (k a constant), the classic simple-harmonic-motion pattern that reappears throughout the chapter whenever a solution of the form A cos(something)+B sin(something) is involved.
3. d²w/dx² − x²(dw/dx) + w = 0 — a second-order equation with variable (not constant) coefficients, using w as the dependent variable instead of y, showing the letter used for the function is not fixed.
4. d²y/dt² + d²x/dt² = x, where BOTH x and y are stated to be functions of the single variable t — a reminder that a differential equation can link two dependent functions of one common independent variable, not just one dependent and one independent variable.
5. d³y/dx³ + x(dy/dx) − 4xy = 0, where x is explicitly said to be a function of y (the reverse of the usual convention) — showing that which variable plays 'independent' and which plays 'dependent' is a matter of the problem's own setup, not a fixed rule.
6. r(dr/dθ) + cos θ = 5 — using polar-style variables r and θ instead of x and y, to emphasise the definition does not depend on any particular choice of variable names. …