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 |
