# Differential Equations Overview

> AP Calculus AB · Differential Equations
> Source: https://www.owlsprep.com/study/ap-calculus-ab-u7-overview/
> Weight: 6-12% of the total AP Calculus AB exam score

This unit introduces first-order differential equations, which relate functions to their derivatives. You will learn to solve, model, and interpret these core equations, including the key application of exponential growth and decay.

**Prerequisites:** [Basic differentiation rules](https://www.owlsprep.com/study/ap-calculus-ab-u3-differentiation-techniques-overview/); [Indefinite integration](https://www.owlsprep.com/study/ap-calculus-ab-u6-indefinite-integration-constant-of-integration/)

## Learning objectives

- Recognize differential equations and interpret their meaning in applied modeling contexts
- Verify solutions to differential equations and use slope fields to analyze solution behavior
- Use separation of variables to find general and particular solutions to separable differential equations
- Model exponential growth and decay with first-order differential equations

## Unit at a Glance

This unit builds from foundational basics to applied problem solving. We start with what a differential equation is and how to verify potential solutions. Next, you will learn to translate real-world scenarios into differential equation models, then explore graphical representations of solutions via slope fields. Finally, we cover the separation of variables technique for finding analytical solutions, ending with the core application of exponential growth and decay modeling.

Sub-topics in this unit are structured to build incrementally, as follows:
- [AP Calculus AB Exponential models with differential equations](https://www.owlsprep.com/study/ap-calculus-ab-u7-exponential-models-with-differential-equations/) — Apply differential equations to model exponential growth and decay in real-world contexts.
- [AP Calculus AB General solutions via separation of variables](https://www.owlsprep.com/study/ap-calculus-ab-u7-general-solutions-via-separation-of/) — Use separation of variables to find families of general solutions for separable differential equations.
- [AP Calculus AB Modeling situations with differential equations](https://www.owlsprep.com/study/ap-calculus-ab-u7-modeling-situations-with-differential-equations/) — Translate word problems and real scenarios into valid differential equation representations.
- [AP Calculus AB Particular solutions with initial conditions](https://www.owlsprep.com/study/ap-calculus-ab-u7-particular-solutions-with-initial-conditions/) — Use initial conditions to solve for the constant of integration and find unique particular solutions.
- [AP Calculus AB Reasoning using slope fields](https://www.owlsprep.com/study/ap-calculus-ab-u7-reasoning-using-slope-fields/) — Interpret slope fields to draw solution curves and predict long-run solution behavior.
- [AP Calculus AB Sketching slope fields](https://www.owlsprep.com/study/ap-calculus-ab-u7-sketching-slope-fields/) — Learn to hand-sketch slope fields for any first-order differential equation.
- [AP Calculus AB Verifying solutions for differential equations](https://www.owlsprep.com/study/ap-calculus-ab-u7-verifying-solutions-for-differential-equations/) — Check if a given function satisfies a differential equation by substitution.

## Common pitfalls

- **Wrong:** Forgetting to add the constant of integration after integrating separated sides.
  - Why it fails: Missing $C$ means you only get one solution instead of the full family of general solutions.
  - Correct: Add $C$ to one side immediately after completing integration of both sides.
- **Wrong:** Making algebra errors when separating terms to opposite sides.
  - Why it fails: Incorrect separation of $x$ and $y$ terms leads to wrong integrals and final solutions.
  - Correct: Double-check your separation step before moving on to integration.
- **Wrong:** Solving for $C$ before writing the general solution.
  - Why it fails: Substituting the initial condition too early leads to incorrect values for the constant.
  - Correct: Always find the general solution first, then substitute the initial condition to solve for $C$.

## Cheatsheet

| Concept | Key Result / Formula |
| --- | --- |
| First-order differential equation | Equation of the form $\frac{dy}{dx} = f(x,y)$ relating $y$ to its first derivative |
| Verifying a solution | Differentiate $y(x)$, substitute $y$ and $\frac{dy}{dx}$ into the DE to confirm equality |
| Slope field rule | Slope at $(x,y)$ equals $\frac{dy}{dx}$ evaluated at that point |
| Separation of variables steps | 1. Separate $x$/$y$ terms 2. Integrate both sides 3. Solve for $y$ |
| General vs particular solution | General: family of solutions with constant $C$; Particular: unique solution matching an initial condition |
| Exponential change differential equation | $\frac{dy}{dt} = ky$, solution: $y(t) = y_0 e^{kt}$ where $y_0 = y(0)$ |
| Initial condition notation | $y(t_0) = y_0$: value of the solution at the starting point $t_0$ |

## What's next

Start this unit with the foundational topic of verifying solutions to build core intuition for what differential equations are. Work through the sub-topics in order to build your skills from basics to applied problem solving. Once you complete all topics in this differential equations unit, you will move on to the next unit covering applications of integration.

- [AP Calculus AB Verifying solutions for differential equations](https://www.owlsprep.com/study/ap-calculus-ab-u7-verifying-solutions-for-differential-equations/)
- [Modeling situations with differential equations](https://www.owlsprep.com/study/ap-calculus-ab-u7-modeling-situations-with-differential-equations/)
- [Sketching Slope Fields](https://www.owlsprep.com/study/ap-calculus-ab-u7-sketching-slope-fields/)

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