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 |