Qualitative Analysis of ODEs
Updated 07 Jul 2021
Created with R2021a. Compatible with R2021a and later releases.
This curriculum module teaches qualitative analysis of ordinary differential equations (ODEs) using interactive live scripts and MATLAB® apps. 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.
An interactive lesson that teaches how to analyze first order ODEs using slope fields.
- Classify ODEs as autonomous or non-autonomous
- Describe how a slope field is generated
- Draw ODE solutions through the slope field by hand
- Explain the physical meaning of terms in an energy balance climate model
- Plot solutions of the energy balance climate model
A live script lesson that explains how to find and classify equilibria of first order, autonomous differential equations.
- Solve for equilibria of autonomous first order ODEs
- Classify equilibria as stable, unstable, or semi-stable
- Draw phase line portraits of autonomous ODEs and relate them to the slope field
- Find the equilibria of the energy balance climate model
- Describe the real-world meaning of the climate model equilibria
In this lesson, students learn to write a second order ODE as a system of two first order ODEs and analyze the system using the phase plane.
- Write a second order ODE as a first order ODE system
- Describe how a phase plane is generated
- Relate solutions plotted on the phase plane to time series solution plots
- Compare behaviors of the mass-spring-damper model under variations in the damping coefficient
An interactive lesson that explains how to find and classify equilibria of two dimensional, linear ODE systems.
- Find equilibria of a second order ODE system
- Compute the trace, determinant, and eigenvalues of the coefficient matrix of a linear ODE system
- Classify equilibria as nodes, saddles, spirals, or centers
- Categorize the behavior of the mass-spring-damper model by analyzing its coefficient matrix
MATLAB, Symbolic Math Toolbox™
The license for this module is available in the LICENSE.md file in this GitHub repository.
Have any questions or feedback? Contact the MathWorks online teaching team.
Copyright 2021 The MathWorks, Inc.
Brian Hong (2021). Qualitative Analysis of ODEs (https://github.com/MathWorks-Teaching-Resources/Qualitative-Analysis-of-ODEs/releases/tag/v1.0.0), GitHub. Retrieved .
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