Unit 7 · Differential Equations
● Core concept · ○ Supporting concept
7.1 Modeling Situations with 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.
7.2 Verifying Solutions for Differential Equations
Solution of a differential equation ● (core concept) — A function that satisfies the differential equation when substituted in, along with its derivatives.
7.3 Sketching Slope Fields
Slope field ● (core concept) — A grid of short line segments showing the slope dy/dx at many points; solution curves follow the segments.
7.5 Approximating Solutions Using Euler's Method
Euler's method ● (core concept) — A numerical procedure for approximating a solution to a differential equation (or a point on a solution curve): from a known point, step forward using the slope given by the equation — y_new = y_old + f(x_old, y_old)·Δx — repeated with a small step size Δx.
7.6 Finding General Solutions Using Separation of Variables
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.
7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables
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.
7.8 Exponential Models with Differential Equations
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).
7.9 Logistic Models with Differential Equations
Logistic growth model ● (core concept) — The differential equation dy/dt = ky(a − y), which arises when the rate of change of a quantity is jointly proportional to the size of the quantity and to the difference between the quantity and its carrying capacity.
Carrying capacity ● (core concept) — The limiting value a in a logistic model: as the independent variable approaches infinity, the dependent variable approaches a. Solutions approach the carrying capacity but never cross it.
Point of fastest logistic growth ● (core concept) — In logistic growth, the quantity changes fastest when it equals half the carrying capacity (y = a/2) — the inflection point of the solution curve. This can be read off the model without solving the equation.