Lecture Notes and Readings
Required readings are listed in the table below. Also included are lecture notes developed by the instructor to supplement the reading assignments. There are no supplementary notes for L15-18 and L31-35.
[BR] = section numbers in Birkhoff, Garret, and Gian-Carlo Rota. Ordinary Differential Equations. 4th ed. New York, NY: Wiley, 1989. ISBN: 9780471860037.
| SES # | TOPICS | LECTURE NOTES | READINGS |
|---|---|---|---|
| L0 | Terminology and implicit solutions | (PDF) | [BR] Sec. 1.1 Terminology & Implicit Solutions |
| Unit I: First-order differential equations | |||
| L1 | Integration and solutions | (PDF) | [BR] Sec. 1.2 Fundamental Theorem of the Calculus & Method by Quadrature |
| L2 | Fundamental principles | (PDF) | [BR] Sec. 1.9 Fundamental Principles - Linearity, Existence and Uniqueness, Stability |
| L3 | First-order linear equations | (PDF) | [BR] Sec. 1.3 First-order Linear Equations & Logarithmic Spirals |
| L4 | Separable equations | (PDF) | [BR] Sec. 1.4 Separable Equations & Orthogonal Trajectories |
| L5 | Linear fractional equations | (PDF) | [BR] Sec. 1.7 Linear Fractional Equations |
| Unit II: Second-order linear equations | |||
| L6 | Second-order linear equations | (PDF) | [BR] Sec. 2.1-2.2 Second-order Linear Equations |
| L7 | Mechanical oscillation | ( PDF) |
|
| L8 | Uniqueness and the wronskian | (PDF) | [BR] Sec. 2.4-2.5 Uniqueness & the Wronskian |
| L9 | Separation and comparison theorems | (PDF) | [BR] Sec. 2.6 Separation and Comparison Theorems |
| L10 | The maximum principle | (PDF) | |
| Unit III: Higher-order linear equations | |||
| L11 | Higher-order linear equations | (PDF) | [BR] Sec. 3.1-3.3 The Characteristic Polynomial |
| L12 | Solution bases | (PDF) | [BR] Sec. 3.4 Solution Bases - Existence & Uniqueness |
| L13 | Inhomogeneous equations | (PDF) | [BR] Sec. 3.5 Inhomogeneous Equations |
| L14 | Stability | (PDF) | [BR] Sec. 2.3, 3.7 Asymptotic Stability |
| L15 | Wellposedness; introduction | [BR] Sec. 6.2, 6.6 Wellposedness; Introduction | |
| L16 | Uniform convergence | [BR] Sec. 6.7, 6.9 Picard's Iteration | |
| L17 | Uniqueness and continuity | [BR] Sec. 6.3, SS1.9-1.10 Uniqueness and Continuity | |
| L18 | Remarks on wellposedness | [BR] Sec. 6.5, 6.8, 6.10 Remarks on Wellposedness | |
| Unit V: The Laplace transform | |||
| L19 | Laplace transform | (PDF) | |
| L20 | Transform and differential equations: generalized solutions, application to ODEs | (PDF) | |
| L21 | Step functions | (PDF) | |
| L22 | Convolution | (PDF) | |
| L23 | The dirac distribution | (PDF) | |
| L24 | The transfer function and the pole diagram | (PDF) | |
| Unit VI: The linear systems | |||
| L25 | Linear systems | (PDF) | [BR] Sec. 5.4 Matrices & Linear Systems |
| L26 | Eigenvalues and eigenvectors | (PDF) | [BR] Sec. 5.4 Eigenvalues & Eigenvectors |
| L27 | Complex solutions and the fundamental matrix | (PDF) | |
| L28 | Repeated eigenvalues and the matrix exponential | (PDF) | [BR] Appendix A1-2 Repeated Eigenvalues & Matrix Exponential |
| L29 | Phase planes I | (PDF) | [BR] Sec. 5.5 Phase Planes |
| L30 | Phase planes II | (PDF) | [BR] Sec. 5.5 Phase Planes; Degenerate cases |
| L31 | Plane autonomous system | [BR] Sec. 5.1-5.2, 5.7 Plane Autonomous System | |
| L32 | Stability and almost linear systems | [BR] Sec. 5.7-5.8 Stability and Almost linear systems | |
| L33 | Problems from ecology | ||
| L34 | Methods of Lyapunov | [BR] Sec. 5.7-5.8 Methods of Lyapunov | |
| L35 | Nonlinear oscillations | [BR] Sec. 5.9-5.11 Nonlinear Oscillations | |
| L36 | The Poincare-Bendixson theorem | (PDF) |
[BR] Sec. 5.12 The Poincare-Bendixson Theorem |
