AP Calculus BC Unit 7: Differential Equations (Slope Fields, Euler’s Method, Separable IVPs, Exponential and Logistic Models)

Differential equation

An equation that relates an unknown function (such as y) to one or more of its derivatives (such as y′).

Dependent variable

The output variable whose value depends on the independent variable (often y in y(x)).

Independent variable

The input variable that the function depends on (often x or t).

Derivative notation (y′)

A common notation meaning the derivative of y with respect to x.

Derivative notation (dy/dx)

Notation for the derivative of y with respect to x; represents a derivative, not a fraction in general.

Order (of a differential equation)

The highest derivative that appears in the differential equation.

First-order differential equation

A differential equation whose highest derivative is first derivative (y′).

Second-order differential equation

A differential equation whose highest derivative is the second derivative (y′′).

Solution (to a differential equation)

A function that makes the differential equation true wherever it is defined.

Verify a solution

Check whether a proposed function satisfies a differential equation by substituting it (and its derivative(s)) into the equation.

General solution

A family of solutions that includes an arbitrary constant (often C).

Constant of integration (C)

An arbitrary constant added after integrating, representing a family of antiderivatives/solutions.

Particular solution

The specific solution obtained after using an initial condition to determine the constant(s).

Initial condition

A given value like y(x0)=y0 used to select a particular solution from a general solution.

Initial value problem (IVP)

A differential equation together with an initial condition (or conditions).

Separation of variables

A solving method that rearranges a first-order DE so all y-terms are on one side and all x-terms are on the other.

Separable differential equation

A first-order DE that can be rewritten in the form dy/dx=f(x)g(y) and separated into (1/g(y))dy=f(x)dx.

Differentials (dx and dy)

Symbols used in calculus that, in separable DEs, allow rewriting dy/dx into a form that can be integrated on each side.

Implicit solution

A solution written as a relationship involving both x and y, not solved completely for y.

Explicit solution

A solution written with y isolated, e.g., y = (some expression in x).

Slope field (direction field)

A diagram of small line segments showing the slope y′ at many points (x,y), representing solution curve directions.

Direction segment

A small line piece in a slope field drawn with slope F(x,y) at a specific point (x,y).

Isocline

A curve in the xy-plane along which the slope y′ is constant; defined by F(x,y)=k.

Autonomous differential equation

A differential equation where y′ depends only on y (e.g., y′=G(y)), not explicitly on x.

Equilibrium solution

A constant solution y=c (or P=c) where the derivative is zero everywhere; occurs when G(c)=0 in y′=G(y).

Stable equilibrium

An equilibrium that nearby solutions move toward as x (or t) increases.

Unstable equilibrium

An equilibrium that nearby solutions move away from as x (or t) increases.

Sign analysis (for y′)

Determining where solutions increase/decrease by checking whether y′ is positive or negative in different regions.

Euler’s method

A numerical method that approximates solutions to an IVP by stepping forward using tangent-line slopes.

Step size (h)

The increment in x (or t) used in Euler’s method; smaller h usually improves accuracy.

Euler update formula

The recursion y{n+1}=yn + h·F(xn,yn), using the slope at the beginning of each step.

Local truncation error

The error made in a single Euler step due to approximating the curve by a tangent line over that step.

Global error

The accumulated error after many Euler steps over an interval.

Tangent line approximation

Using the tangent line at a point to approximate the function’s value a short distance away (the idea behind Euler’s method).

Concavity heuristic (Euler under/overestimate)

A rule of thumb: if the true solution is concave up, Euler’s method often underestimates; if concave down, it often overestimates.

Exponential growth

Growth modeled by dy/dt=ky with k>0, giving solutions of the form y=Ae^{kt}.

Exponential decay

Decay modeled by dy/dt=ky with k<0, giving solutions of the form y=Ae^{kt}.

Proportionality constant (k)

The constant in dy/dt=ky that controls growth/decay rate; has units of “per unit time” if t is time.

Doubling time

For dy/dt=ky with k>0, the time T for a quantity to double: T=(ln 2)/k.

Half-life

For dy/dt=ky with k<0, the time T for a quantity to halve: T=ln(1/2)/k (a positive value for decay).

Exponential model solution y=y0e^{kt}

The IVP solution to dy/dt=ky with y(0)=y0.

Logistic differential equation

A growth model dP/dt=rP(1−P/K) that slows as P approaches the carrying capacity K.

Carrying capacity (K)

In the logistic model, the maximum sustainable population; the stable long-term level when r>0.

Intrinsic growth rate (r)

In the logistic model, a constant controlling how quickly the population grows when far below carrying capacity; units of “per unit time.”

Partial fractions (logistic integration)

An algebra technique used to integrate 1/[P(K−P)] when solving the logistic equation by separation.

Logistic solution form P=K/(1+Be^{−rt})

A common explicit solution to the logistic DE, where B is determined by the initial condition.

Inflection point (logistic curve)

The point where logistic growth is fastest and concavity changes; occurs at P=K/2.

Units consistency check

Using units to verify a model: dy/dt must have the same units as the right-hand side; constants must carry appropriate units.

Interpretation of y′(a)=F(a,b)

If a solution passes through (a,b), then y′(a)=F(a,b), giving the instantaneous rate of change at that point (with contextual units).

Chain rule in y′′ from y′=F(x,y)

To find concavity, differentiate y′=F(x,y); if F involves y, include a y′ term via the chain rule (e.g., d/dx(−y)=−y′).