Unit 7 · Differential Equations

Differential equation ● (core concept) — An equation relating a function to its derivatives, e.g., dy/dx = ky; it models how a quantity changes.

Solution of a differential equation ● (core concept) — A function that satisfies the differential equation when substituted in, along with its derivatives.

Slope field ● (core concept) — A grid of short line segments showing the slope dy/dx at many points; solution curves follow the segments.

Separation of variables ● (core concept) — Solving dy/dx = g(x)·h(y) by rewriting as (1/h(y)) dy = g(x) dx and integrating both sides.

General solution ● (core concept) — The family of all solutions of a differential equation, containing an arbitrary constant C.

Particular solution ● (core concept) — The single solution of a differential equation that also satisfies a given initial condition (no arbitrary constant remains).

Initial condition ● (core concept) — A specified value of the solution, e.g., y(0) = 3, used to determine the constant C in the general solution.

Exponential model ● (core concept) — A model in which a quantity's rate of change is proportional to the quantity itself: dy/dt = ky, with solutions y = Ce^(kt) (growth if k > 0, decay if k < 0).