Differential Equations

Qualitative Behavior: Phase Portraits

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In this session we will leave off looking for exact solutions to constant coefficient systems of DE’s and focus on the qualitative features of the solutions. The main tool will be phase portraits, which are sketches of the trajectories of solutions in the xy-plane (now called the phase plane). We will see that the qualitative nature of the solutions is determined by the eigenvalues of the coefficient matrix.

Session Activities

Read the course notes:

• Qualitive Behavior: Phase Portraits: Introduction (PDF)

Watch the lecture video clips:

• Sketching Solutions
• More Sketching Solutions

Read the course notes:

• The Phase Plane (PDF)
• Sketching the Basic Linear Systems (PDF)
• Sketching More General Linear Systems (PDF)
• Summary (PDF)
• Trace-Determinant Diagram (PDF)
• Appendix: A Computer-Generated Portrait Gallery (PDF)

Learn from the Mathlet materials:

• Read about how to work with the Linear Phase Portraits: Cursor Entry Applet (PDF)
• Work with the Linear Phase Portraits: Cursor Entry Applet

Watch the problem solving videos:

• Trace-Determinant Diagram
• Phase Portraits

Complete the practice problems:

• Practice Problems 34 (PDF)
• Practice Problems 34 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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