---
title: Second-Order ODEs
description: Second-order ordinary differential equations involve derivatives up to the second degree. Learners will solve linear homogeneous and non-homogeneous equations, applying these techniques to model physical phenomena like mechanical vibrations and electrical circuits.
category: mathematics
subcategory: differential-equations
difficulty: beginner, intermediate, advanced
url: /subject/second-order-odes
---

# Second-Order ODEs

Second-order ordinary differential equations involve derivatives up to the second degree. Learners will solve linear homogeneous and non-homogeneous equations, applying these techniques to model physical phenomena like mechanical vibrations and electrical circuits.

## Available Resources

1 Books • 2 Courses • 3 Websites

## Websites

### 1. Khan Academy Differential Equations

Khan Academy unit of short videos on linear second-order differential equations, covering the characteristic equation with real, complex, and repeated roots, and the method of undetermined coefficients. Learners can solve homogeneous and nonhomogeneous constant-coefficient equations.

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

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

**Tags:** second-order-odes, differential-equations, characteristic-equation, undetermined-coefficients

### 2. MIT Mathlets

**Author:** Haynes Miller

Browser-based applets from MIT's 18.03 course. Damped Vibrations, Amplitude and Phase: Second Order, Forced Damped Vibration, and Harmonic Frequency Response let you move characteristic roots, damping, and driving frequency and watch solutions respond.

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

**Link:** https://mathlets.org/mathlets/

**Tags:** differential-equations, second-order-odes, damped-oscillations, frequency-response, interactive-applets

### 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. Notes on Diffy Qs: Differential Equations for Engineers

**Author:** Jiří Lebl

Free open textbook by an Oklahoma State mathematician. Chapter 2 treats higher-order linear ODEs: constant-coefficient solutions, undetermined coefficients, variation of parameters, mechanical vibrations, and forced oscillations with resonance, each with exercises, solutions, and Sage animations.

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

**Link:** https://www.jirka.org/diffyqs/

**Tags:** differential-equations, second-order-odes, linear-odes, mechanical-vibrations, open-textbook

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