Differential Equations

Geometric Methods

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Session Overview

In this session we will look at graphical methods for visualizing DE’s and their solutions. The primary tool for doing this will be the direction field. We will learn techniques to sketch this by hand and also learn to use direction fields drawn by the computer.

Session Activities

Read the course notes:

• Geometric Methods: Introduction (PDF)

Watch the lecture video clip:

• Geometric View of DE’s: Direction Fields, Integral Curves

Read the course notes:

• Direction Fields, Isoclines, Integral Curves (PDF)

Watch the lecture video clip:

• Example, Existence and Uniqueness

Complete the practice problem:

• Isoclines (PDF)
• Solution (PDF)

Session Activities

Read the course notes:

• Existence and Uniqueness Theorem for ODE’s (PDF)

Watch a lecture video clip:

• Example Where Existence and Uniqueness Fails

Learn from the Mathlet materials:

• Watch the video Exploration of the Isoclines Applet
• Read about how to work with the Isoclines Applet (PDF)
• Work with the Isoclines Applet

Read the course notes:

• Long-Term Behavior: Fences, Funnels and Separatrices (PDF)

Watch the problem solving video:

• Direction Fields

Complete the practice problems:

• Practice Problems (PDF)
• Practice Problems Solutions (PDF)

Check Yourself

Complete the problem sets:

• Problem Set Part I Problems (PDF)

• Problem Set Part I Solutions (PDF)

• Problem Set Part II Problems (PDF)

• Problem Set Part II Solutions (PDF)

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