---
title: First-Order ODEs
description: First-order ordinary differential equations involve only the first derivative of an unknown function. Learners will master techniques like separation of variables, integrating factors, and exact equations to solve basic dynamical models.
category: mathematics
subcategory: differential-equations
difficulty: beginner, intermediate, advanced
url: /subject/first-order-odes
---

# First-Order ODEs

First-order ordinary differential equations involve only the first derivative of an unknown function. Learners will master techniques like separation of variables, integrating factors, and exact equations to solve basic dynamical models.

## Available Resources

1 Books • 2 Courses • 3 Websites

## Websites

### 1. Khan Academy Differential Equations

Khan Academy unit of short videos and practice exercises covering slope fields, separable and exact equations, integrating factors, and homogeneous equations. Learners finish able to classify a first-order ODE and solve it with the matching standard technique.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://www.khanacademy.org/math/differential-equations/first-order-differential-equations

**Tags:** first-order-odes, separable-equations, exact-equations, integrating-factors, slope-fields

### 2. Paul's Online Math Notes - Differential Equations

Paul Dawkins' free Lamar University notes for a first ODE course, with worked examples on first-order equations, second-order linear equations, Laplace transforms, systems and series solutions, so learners can solve initial value problems by hand step by step.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://tutorial.math.lamar.edu/Classes/DE/DE.aspx

**Tags:** differential-equations, ordinary-differential-equations, initial-value-problems, laplace-transforms

### 3. Wolfram MathWorld

**Author:** Eric W. Weisstein

MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://mathworld.wolfram.com

**Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry

## Courses

### 1. Introduction to Differential Equations

Learn differential equations: explore methods for solving, modeling, and analyzing them. Taught by the Khan Academy Team.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://www.khanacademy.org/math/differential-equations

**Tags:** differential-equations, ordinary-differential-equations, separable-equations, laplace-transform, khan-academy

### 2. Differential Equations for Engineers

This course is all about differential equations and covers both theory and applications. In the first five weeks, students will learn about ordinary differential equations, while the sixth week is an introduction to partial differential equations.

The course includes 56 concise lecture videos, with a few problems to solve after each lecture. After each major topic, there is a short practice quiz. At the end of each week, there is an assessed quiz. Solutions to the problems and practice quizzes can be found in the instructor-provided lecture notes. 

Download the lecture notes from the link
https://www.math.hkust.edu.hk/~machas/differential-equations-for-engineers.pdf

Watch the promotional video from the link
https://youtu.be/eSty7oo09ZI

**Difficulty:** Beginner | **Language:** English | **Duration:** 6 weeks of study, 4 hours/week | **Price:** Free

**Link:** https://www.coursera.org/learn/differential-equations-engineers

**Tags:** courses, mathematics-statistics, calculus

## Books

### 1. Differential Equations: An Introduction to Modern Methods

**Author:** James R. Brannan, William E. Boyce

Undergraduate textbook by Brannan and Boyce that treats differential equations through a systems and modelling lens, mixing analytic, geometric, and numerical methods. Readers come away able to solve first-order and linear equations and analyse phase portraits of linear and nonlinear systems.

**Difficulty:** Intermediate | **Language:** English | **Price:** Paid

**Link:** https://www.amazon.com/dp/0470458240?tag=edmonddante07-20

**Tags:** books, mathematics-statistics, differential-equations

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