AP Calculus AB Unit 7: Differential Equations Notes

Introduction

Differential equations define functions through rates rather than explicit formulas. Models must preserve variables, signs, parameters, units, and domains.

Slope fields display local directions and support qualitative reasoning before a solution is found.

Separation solves multiplicative first-order equations; initial conditions select one family member, and proportional-rate laws produce exponential models.

Current exam guidance: May 2027 format: 42 multiple-choice questions and 6 free-response questions. Calculator availability changes by part; the reader marks calculator decisions separately from mathematical justification.

7.1

Modeling Situations with Differential Equations

How does a verbal change law become a differential equation?

Short answer

Place the requested rate on the left and express it in terms of the current variables and correctly signed parameters.

Know firstDerivative meaning · Proportionality · Units

Write the rate relationship before solving

A differential equation specifies how an unknown function changes. “Rate proportional to amount” becomes y′=ky, while an added inflow or removal term changes the right side.

Define variables and units first. The sign of each process matters, and an equilibrium occurs where the modeled rate is zero.

Formula

Formula and conditions

dy/dt=F(t,y)

The right side states the current rate law.

Use when: Variables, parameters, units, and domain match the context.
Do not assume: Writing y rather than y′ models amount, not change.

Current amount controls proportional rate

For y′=0.4y, larger positive y-values have steeper positive slopes; y=0 is an equilibrium.

The interactive version of this figure is part of the full study guide.

Current amount controls proportional rate: values

Case	Result
Series 1	$y=2e^{0.4x}$

Reason from the differential equation

Define variables.
Translate each rate phrase.
Attach parameter signs and units.
Identify domain and equilibria.

Keep these three ideas

Rate law first.
Signs encode direction.
Equilibria make the rate zero.

Keep reading Unit 7: Differential Equations

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7.1 Modeling Situations with Differential EquationsRead

7.2 Verifying Solutions for Differential Equations

7.3 Sketching Slope Fields

7.4 Reasoning Using Slope Fields

7.5 Finding General Solutions Using Separation of Variables

7.6 Finding Particular Solutions Using Initial Conditions and Separation of Variables

7.7 Exponential Models with Differential Equations

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