Qualitative Analysis of ODEs

or

Curriculum Module

Created with R2021a. Compatible with R2024a and later releases.

Information

This curriculum module contains interactive MATLAB® live scripts that teaches qualitative analysis of ordinary differential equations (ODEs).

Background

You can use these live scripts as demonstrations in lectures, class activities, or interactive assignments outside class. The first two scripts consider first order ODEs. In these scripts, students draw solutions on a slope field, identify and classify equilibria, and draw phase line portraits. In the second half of the module, students analyze systems of two autonomous ODEs. In this part, students create phase plane portraits, solve for equilibria, and classify equilibria of linear systems by examining the eigenvalues of the coefficient matrix. Throughout the module, students apply the mathematical tools to study an energy balance climate model and a mass-spring-damper model.

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 knowledge of differential calculus concepts and notation: both $y^{\prime }$ and $\frac{dy}{dt}$ are used. There is minimal MATLAB knowledge required for these scripts, but you could use MATLAB Onramp as a resource 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 ODE_Analysis.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® and Symbolic Math Toolbox™ are used throughout.

Scripts

SlopeFields.mlx

	In this script, students will...	Applications
	$\bullet$ classify ODEs as autonomous or non-autonomous $\bullet$ describe how a slope field is generated $\bullet$ draw ODE solutions through the slope field by hand $\bullet$ explain the physical meaning of terms in an energy balance climate model $\bullet$ plot solutions of the energy balance climate model	Energy balance model

Equilibria1D.mlx

	In this script, students will...	Application
	$\bullet$ solve for equilibria of autonomous first order ODEs $\bullet$ classify equilibria as stable, unstable, or semi-stable $\bullet$ draw phase line portraits of autonomous ODEs and relate them to the slope field $\bullet$ find the equilibria of the energy balance climate model $\bullet$ describe the real-world meaning of the climate model equilibria	Energy balance model

PhasePlanes.mlx

	In this script, students will...	Application
	$\bullet$ write a second order ODE as a first order ODE system $\bullet$ describe how a phase plane is generated $\bullet$ relate solutions plotted on the phase plane to time series solution plots $\bullet$ compare behaviors of the mass-spring-damper model under variations in the damping coefficient

LinearPhasePlaneAnalysis.mlx

	In this script, students will...	Application
	$\bullet$ find equilibria of a second order ODE system $\bullet$ compute the eigenvalues of the coefficient matrix of a linear ODE system $\bullet$ classify equilibria as nodes, saddles, spirals, or centers $\bullet$ categorize the behavior of the mass-spring-damper model by analyzing its coefficient matrix

Related Courseware Modules

Phase Plane and Slope Field

	Available on: GitHub

Or feel free to explore our other modular courseware content.

Related Self-Paced Online Courses

Solving Ordinary Differential Equations with MATLAB

Learn the basics of solving ordinary differential equations in MATLAB. Use MATLAB ODE solvers to find solutions to ordinary differential equations that describe phenomena ranging from population dynamics to the evolution of the universe.

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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