Applied Ordinary Differential Equations

or

Curriculum Module

Created with R2024b. Compatible with R2024b and later releases.

Information

This curriculum module contains interactive MATLAB® live scripts that cover analytical methods of solving ordinary differential equations, using applications in a variety of academic disciplines.

Background

You can use these live scripts as demonstrations in lectures, class activities, or interactive assignments outside class. This module covers core topics from an introductory differential equations course. This includes solving first-order equations using separation of variables and integrating factors, solving second-order equations using characteristic equations and the method of undetermined coefficients, and systems of differential equations by finding eigenvalues and eigenvectors. Applications of differential equations include population growth, circuits, measuring chemical concentration, and Newton's law of cooling.

The instructions inside the live scripts will guide you through the exercises and activities. Get started with each live script by running it one section at a time. To stop running the script or a section midway (for example, when an animation is in progress), use the Stop button in the RUN section of the Live Editor tab in the MATLAB Toolstrip.

Contact Us

Solutions are available upon instructor request. Contact the MathWorks teaching resources team if you would like to request solutions, provide feedback, or if you have a question.

Prerequisites

This module assumes a background in calculus (taking derivatives and integrals), which are covered by related courseware modules. Minimal prior MATLAB experience required, as the focus of the scripts are mostly analytical techniques, but MATLAB Onramp can be used to acquire familiarity with live scripts and MATLAB syntax.

Getting Started

Accessing the Module

On MATLAB Online:

Use the link to download the module. You will be prompted to log in or create a MathWorks account. The project will be loaded, and you will see an app with several navigation options to get you started.

On Desktop:

Download or clone this repository. Open MATLAB, navigate to the folder containing these scripts and double-click on AppliedODEs.prj. It will add the appropriate files to your MATLAB path and open an app that asks you where you would like to start.

Ensure you have all the required products (listed below) installed. If you need to include a product, add it using the Add-On Explorer. To install an add-on, go to the Home tab and select Add-Ons > Get Add-Ons.

Products

MATLAB is used throughout, and tools from the Symbolic Math Toolbox™ are used frequently as well.

Scripts

Classification.mlx

	In this script, students will... $\bullet$ identify independent and dependent variables in differential equations $\bullet$ distinguish between ordinary and partial differential equations $\bullet$ classify differential equations by order, linearity, and homogeneity	Academic disciplines $\bullet$ Mathematics

SeparationOfVariables.mlx

	In this script, students will... $\bullet$ determine whether ordinary differential equations are separable $\bullet$ solve first-order ordinary differential equations analytically using separation of variables	Applications $\bullet$ Logistic population growth model Academic disciplines $\bullet$ Mathematics $\bullet$ Biology

IntegratingFactors.mlx

	In this script, students will... $\bullet$ find integrating factors of linear, first-order ordinary differential equations $\bullet$ solve equations analytically by multiplying by integrating factors	Applications $\bullet$ Chemical concentration Academic disciplines $\bullet$ Mathematics $\bullet$ Chemistry

CharacteristicEquations.mlx

	In this script, students will... $\bullet$ find characteristic equations of linear, second-order ordinary differential equations $\bullet$ use the roots of characteristic equations to solve homogeneous equations analytically	Applications $\bullet$ RLC Circuits $\bullet$ Mass-spring-damper Academic disciplines $\bullet$ Mathematics $\bullet$ Engineering $\bullet$ Physics

UndeterminedCoefficients.mlx

	In this script, students will... $\bullet$ solve linear, second-order, nonhomogeneous ordinary differential equations analytically using the method of undetermined coefficients	Applications $\bullet$ Pendulum motion $\bullet$ Chemical bonds Academic disciplines $\bullet$ Mathematics $\bullet$ Physics $\bullet$ Chemistry

SystemsOfODEs.mlx

	In this script, students will... $\bullet$ write systems of linear, homogeneous, first-order differential equations in matrix form $\bullet$ find eigenvalues and eigenvectors of these matrices $\bullet$ solve systems of ODEs analytically using these eigenvalues and eigenvectors	Applications $\bullet$ SIR model of epidemiology $\bullet$ Earthquake effects on buildings Academic disciplines $\bullet$ Mathematics $\bullet$ Biology $\bullet$ Engineering

License

The license for this module is available in the LICENSE.md.

Related Courseware Modules

Applied Partial Differential Equations

	Available on: GitHub

Calculus Derivatives

	Available on: GitHub

Calculus Integrals

	Available on: GitHub

Numerical Methods with Applications

	Available on: GitHub

Qualitative Analysis of ODEs

	Available on: GitHub

Or feel free to explore our other modular courseware content.

Educator Resources

• Educator Page

Contribute

Looking for more? Find an issue? Have a suggestion? Please contact the MathWorks teaching resources team. If you want to contribute directly to this project, you can find information about how to do so in the CONTRIBUTING.md page on GitHub.

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