18.034 | Spring 2009 | Undergraduate

Honors Differential Equations

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.

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

Course Info

As Taught In
Spring 2009
Learning Resource Types
Problem Sets with Solutions
Lecture Notes
Projects with Examples