Differential Equations

Modes and the Characteristic Equation

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Session Overview

We now begin an in depth study of constant coefficient linear equations. These are the most important DE’s in 18.03, and we will be studying them up to the last few sessions. In this session we will learn algebraic techniques for solving these equations. Exponential functions will play a major role and we will see that higher order linear constant coefficient DE’s are similar in many ways to the first order equation x’ + kx = 0.

Session Activities

Read the course notes:

• Modes and the Characteristic Equation: Introduction (PDF)

Watch the lecture video clips:

• Constant Coefficient Second Order Homogeneous DE’s
• Real Roots: Overdamping

Read the course notes:

• Second Order Physical Systems (PDF)
• Linear Differential Equations (PDF)

Check Yourself

Take the quiz:

• Period of the Simple Harmonic Oscillator (PDF)
• Choices (PDF)
• Answer (PDF)

Session Activities

Read the course notes:

• The Characteristic Polynomial (PDF)

Watch the lecture video clips:

• Complex Roots: Underdamping
• Repeated Roots: Critical Damping

Read the course notes:

• Modes and Roots (PDF)

Watch the lecture video clip:

• Real Solutions from Complex Roots

Watch the problem solving videos:

• Homogeneous Constant Coefficient Equations: Real Roots
• Homogeneous Constant Coefficient Equations: Any Roots

Complete the practice problems:

• Practice Problems 9 (PDF)
• Practice Problems 9 Solutions (PDF)

Check Yourself

Complete the problem set:

• Problem Set Part I Problems (PDF)
• Problem Set Part I Solutions (PDF)

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