Brief notes on Introduction to Differential Equations

These are brief lecture notes for a course on Introduction to Differential Equations developed when I (Gautam Iyer) taught it at Carnegie Mellon University. These notes intentionally only list definitions, and statements of results, and problems, and occasionally brief sketches of proofs. I do motivation, intuition, and complete proofs in class. There are also several comprehensive references for this material so I don’t plan to include them in these notes.

Contents

1. Introduction to ODEs
   1. Examples of ODEs
   2. Autonomous Linear Equations
   3. Time inhomogeneous linear equations
2. Existence, Uniqueness, etc.
   1. Existence and Uniqueness
   2. Continuous Dependence
   3. Extensibility
   4. Beyond Locally Lipschitz coefficients
3. Lyapunov functions and Stability
   1. Autonomous Equations
   2. Lyapunov Functions
   3. Linearized Stability
   4. Instability
4. A few model PDEs
   1. Transport Equation
   2. Heat equation
   3. Wave Equation
   4. Poisson's equation
   5. Harmonic Functions
   6. Minimal surfaces
5. Method of Characteristics
   1. The characteristic equation
   2. The boundary value problem
   3. General existence theorems
6. Harmonic functions
   1. Newton potentials
   2. Mean value property
   3. Maximum principle
   4. Regularity
   5. Liouville's theorem
   6. Harnack inequality
   7. Energy methods
7. Greens functions and Existence
   1. Greens functions
   2. Poisson formula in upper half space
   3. Poisson formula in a ball
   4. Existence (Perron's method)
8. Elliptic equations
   1. Variable coefficient elliptic operators
   2. Maximum principles
   3. Interior regularity: energy methods
9. The heat equation
   1. Fundamental solution in $\R^d$
   2. Mean value property
   3. Strong Maximum principle
   4. Interior regularity
   5. Energy methods
10. Separation of Variables
    1. Series representation of solutions
    2. Complex Fourier Series

Source Code

The source code is available on Codeberg.

The notes are written in Markdown, and converted to HTML5 using the Python script md-to-html. It should be viewable on most modern browsers, but I only test it on Firefox.

Licence

Brief notes on Introduction to Differential Equations © 2026 by Gautam Iyer is licensed under Creative Commons Attribution-NonCommercial 4.0 International .

You may adapt and share this work for non-commercial purposes, provided you cite it appropriately. The full terms are in the licence deed.