AP Calculus AB Unit 7 Study Guide: Differential Equations (Slope Fields, Separable IVPs, Euler’s Method, Exponential & Logistic Models)

Differential equation

An equation that relates an unknown function (such as y as a function of x) to one or more of its derivatives (such as dy/dx).

Rate of change

A derivative value that describes how fast a quantity is changing with respect to another variable.

Derivative notation (dy/dx)

The derivative of y with respect to x; a common way to write y′.

Prime notation (y′)

Another common notation for dy/dx, the derivative of y with respect to x.

Second derivative (d²y/dx² or y′′)

The derivative of the derivative; describes how the slope dy/dx changes with x.

Explicit function

A function given directly in the form y = f(x), as opposed to being defined by a rule for its derivative.

Differential equation model

A situation where you know (or assume) a rule for how a quantity changes and use it to describe the quantity via an equation involving derivatives.

Solution (to a differential equation)

A function y = f(x) that makes the differential equation true when substituted along with its derivative(s).

General solution

A family of solutions written with an arbitrary constant (often C) to represent infinitely many solutions.

Particular solution

The single solution obtained after using an initial condition to determine the constant in the general solution.

Initial condition

A given value like y(x0) = y0 used to select one specific solution from a family of solutions.

Initial value problem (IVP)

A differential equation together with an initial condition, typically written dy/dx = g(x,y) and y(x0)=y0.

Family of solutions

The infinitely many functions that satisfy a differential equation before an initial condition is applied.

Verify a solution

Check a proposed y by computing required derivatives, substituting into the differential equation, and confirming both sides match for all x in the domain.

Slope field

A diagram showing short line segments whose slopes equal dy/dx at many points (x,y), without solving the differential equation.

Direction field

Another name for a slope field: it displays the direction (slope) of solution curves at points in the plane.

Solution curve (integral curve)

A smooth curve that follows a slope field by staying tangent to its small direction segments at every point.

Plug-and-draw (slope field construction)

Method for building a slope field by plugging each point’s coordinates into g(x,y) and drawing a short segment with that slope.

Horizontal segment in a slope field

Indicates dy/dx = 0 at that point, so solution curves have a horizontal tangent there.

Isocline

A curve where the slope field has constant slope k; it is the set of points satisfying g(x,y)=k.

Equilibrium solution

A constant solution y = a for which dy/dx = 0, so the solution stays constant as x changes.

Constant solution

Another name for an equilibrium solution: y does not change because its derivative is zero.

Autonomous differential equation

A differential equation of the form dy/dx = f(y) that depends only on y (not explicitly on x).

Stable equilibrium

An equilibrium y=a where nearby solutions move toward y=a over time (from above and below).

Unstable equilibrium

An equilibrium y=a where nearby solutions move away from y=a over time.

Stability sign test

For dy/dx=f(y): if f(y)>0 just below a and f(y)<0 just above a, then y=a is stable; reversed signs indicate unstable.

Existence and uniqueness (concept)

If g(x,y) and ∂g/∂y are continuous near (x0,y0), then the IVP typically has a unique solution through that point.

Euler’s method

A numerical method for approximating solutions to an IVP by repeatedly stepping along tangent lines using dy/dx = g(x,y).

Step size (h)

The x-increment used in Euler’s method; smaller h usually improves accuracy but requires more steps.

Euler update rule

Given (xn,yn), compute xn+1=xn+h and yn+1=yn+h·g(xn,yn).

Local linearization

Approximating a function near a point by its tangent line; Euler’s method applies this idea repeatedly.

Euler error accumulation

Euler’s method can drift because each step is approximate and later slopes are computed from earlier approximations.

Separable differential equation

A differential equation that can be rearranged so all y-terms are on one side and all x-terms are on the other, enabling integration.

Separation of variables

Rewriting dy/dx = f(x)g(y) as (1/g(y))dy = f(x)dx and then integrating both sides.

SIPPY checklist

A solving routine for separable DEs: Separate, Integrate, Plus C, Plug in initial condition, Y equals (solve for y).

Constant of integration (+C)

The arbitrary constant added after integrating; essential for capturing the full family of solutions and fitting initial conditions.

Absolute value in ln|y|

When integrating 1/y, the correct antiderivative is ln|y|, not ln(y), because y may be negative.

Implicit solution

A solution left in a form where y is not isolated (e.g., y + ln|y| = x² + C), which can still be a valid final answer.

Lost solution (by dividing)

A common pitfall: dividing by an expression like y can remove constant/equilibrium solutions such as y=0 unless noted separately.

Proportional to (modeling cue)

Phrase indicating a rate is k times a quantity, leading to a model like dy/dt = ky.

Constant of proportionality (k)

The constant multiplier in proportional-rate models; its sign indicates growth (k>0) or decay (k<0).

Exponential growth/decay differential equation

The model dy/dt = ky, expressing that the rate of change is proportional to the amount present.

Exponential solution form

Solving dy/dt=ky gives y=Ae^{kt}; with y(0)=y0, it becomes y=y0e^{kt}.

Doubling time

For y=y0e^{kt} with k>0, the constant time T satisfying 2=e^{kT}, so T=(ln 2)/k.

Half-life

For exponential decay (k<0), the constant time T satisfying 1/2=e^{kT}, so T=ln(1/2)/k (a positive value).

Constant-rate model

A model with dy/dt = k (constant), which produces a linear function y=kt+C, not exponential behavior.

Logistic growth differential equation

A population model dP/dt = rP(1 − P/K) that includes limiting resources via the carrying capacity K.

Carrying capacity (K)

The maximum sustainable population level in the logistic model; P=K is an equilibrium that solutions tend to approach.

Logistic explicit solution form

A common closed-form solution: P(t)=K/(1+Ae^{−rt}), where A is set by the initial condition.

Maximum logistic growth rate (at K/2)

In logistic growth, dP/dt is largest when the population is P = K/2.