18.034 | Spring 2004 | Undergraduate
Honors Differential Equations

## Lecture Notes

The lecture notes below represent a summary of the topics discussed and analyzed in class.

SES # TOPICS
Unit I: First-order ODE’s
1 Modeling and Terminology (PDF)
2 Linear Differential Equations (PDF)
3 Existence and Uniqueness of Solutions: Uniqueness (PDF)
4 Existence and Uniqueness of Solutions: Picard Iterates (PDF)
5 Extension of Solutions (PDF)
6 Slope Fields

Separable Equations

7 Qualitative Analysis (PDF)
8 Approximate Numerical Solutions (PDF)
9 Extra Topics and/or Review
10 In-class Exam 1
Unit II: Second-order ODE’s
11

Homogeneous 2nd Order Linear ODE’s with Constant Coefficients (PDF)

Some Instructions on Plotting Functions in MATLAB® (PDF)

12 Direction Fields

Brief Review of Complex Numbers

13 Inhomogeneous 2nd Order Linear ODE’s (PDF)
14 Theory of 2nd Order Linear and Nonlinear ODE’s (PDF)
15 Beats, Resonance, and Frequency Response Modeling
16 Extra Topics (PDF)
Unit III: Fourier Series
17 Fourier Trigonometric Series (PDF)
18 Half-range and Exponential Fourier Series
19 In-class Exam 2
20 The Dirac Delta Function (PDF)
Unit IV: The Laplace Transform
21 The Laplace Transform: Solving IVP’s (PDF)
22 Properties of the Transform (PDF)
23 Convolution (PDF)
24 Extra Topics (PDF)
Unit V: Linear Systems of ODE’s
25 Compartment Models and Introduction to Linear Algebra (PDF)
26 Eigenvalues, Eigenvectors and Eigenspaces (PDF)
27 Homogeneous Linear Systems: Real Eigenvalues Case (PDF)
28 Homogeneous Linear Systems: Complex Eigenvalues Case
29 Inhomogeneous Linear Systems: Exponentials of Matrices
30 Theory of General Linear Systems of ODE’s (PDF)
31

Extra Topics and/or Review

Supplementary Notes on Jordan Normal Form (PDF)

32 In-class Exam 3
Unit VI: Nonlinear Systems of ODE’s
33 The Fundamental Theorem (PDF)
34 Autonomous Systems and Interacting Species Models (PDF)
35 Stability of Linear and Nonlinear Autonomous Systems (PDF)
36 Conservative Systems and Lyapunov Functions (PDF)
37 Limit Cycles and Planar Autonomous Systems
38 Extra Topics
39 Review
40 Final Exam