Second Course in Elementary Differential Equations, by Paul Waltman, John Wiley & Sons (1986).

Copies of the classnotes are on the internet in PDF format as given below. These notes and supplements have not been classroom tested (and so may have some typographical errors). They are based on a section of Applied Math 1 (MATH 5610) I taught in fall 1996.

Syllabus for the Class. PDF.

Chapter 1. Systems of Linear Differential Equations.

• Section 1.1. Introduction. PDF.
• Section 1.2. Some Elementary Matrix Algebra. PDF.
• Supplement. Proofs of Theorems in Section 1.2. PDF (prepared in Beamer).
• Supplement. Printout of the Proofs of Theorems in Section 1.2. PDF.
• Supplement. Selected Solutions for Section 1.2. PDF.
• Section 1.3. The Structure of Solutions of Homogeneous Linear Systems. PDF.
• Supplement. Proofs of Theorems in Section 1.3. PDF (prepared in Beamer).
• Supplement. Printout of the Proofs of Theorems in Section 1.3. PDF.
• Section 1.4. Matrix Analysis and the Matrix Exponential. PDF.
• Supplement. Proofs of Theorems in Section 1.4. PDF (prepared in Beamer).
• Supplement. Printout of the Proofs of Theorems in Section 1.4. PDF.
• Supplement. Selected Solutions for Sections 1.3 and 1.4. PDF.
• Section 1.5. The Constant Coefficient Case: Real and Distinct Eigenvalues. PDF.
• Supplement. Proofs of Theorems in Section 1.5. PDF (prepared in Beamer).
• Supplement. Printout of the Proofs of Theorems in Section 1.5. PDF.
• Section 1.6. The Constant Coefficient Case: Complex and Distinct Eigenvalues. PDF.
• Supplement. Proofs of Theorems in Section 1.6. PDF (prepared in Beamer).
• Supplement. Printout of the Proofs of Theorems in Section 1.6. PDF.
• Section 1.7. The Constant Coefficient Case: The Putzer Algorithm. PDF.
• Supplement. Proofs of Theorems in Section 1.7. PDF (prepared in Beamer).
• Supplement. Printout of the Proofs of Theorems in Section 1.7. PDF.
• Section 1.8. General Linear Systems. PDF.
• Supplement. Proofs of Theorems in Section 1.8. PDF (prepared in Beamer).
• Supplement. Printout of the Proofs of Theorems in Section 1.8. PDF.
• Section 1.9. Some Elementary Stability Considerations.
• Section 1.10. Periodic Coefficients.
• Section 1.11. Scalar Equations.
• Section 1.12. An Application: Coupled Oscillators.

Test 1 (1.1-1.8). PDF.

Chapter 2. Two-Dimensional Autonomous Systems.

• Section 2.1. Introduction. PDF.
• Section 2.2. The Phase Plane. PDF.
• Supplement. Proofs of Theorems in Section 2.2. PDF (prepared in Beamer).
• Supplement. Printout of the Proofs of Theorems in Section 2.2. PDF.
• Section 2.3. Critical Points of some Special Linear Systems. PDF.
• Section 2.4. Critical Points of General Two-Dimensional Linear Systems. PDF.
• Supplement. Proofs of Theorems in Section 2.4. PDF (prepared in Beamer).
• Supplement. Printout of the Proofs of Theorems in Section 2.4. PDF.
• Section 2.5. Behavior of Nonlinear Two-Dimensional Systems Near a Critical Point. PDF.
• Section 2.6. Elementary Liapunov Stability Theory.
• Supplement. Liapunov Functions. PDF.
• Section 2.7. Limit Cycles and the Poincare-Bendixson Theorem.
• Section 2.8. An Example: Lotka-Volterra Competition.
• Section 2.9. An Example: The Simple Pendumlum.

Test 2 (2.1-2.6). PDF.

Chapter 3. Existence Theory.

• Section 3.1. Introduction. PDF.
• Section 3.2. Preliminaries. PDF.
• Supplement. Proofs of Theorems in Section 3.2. PDF (prepared in Beamer).
• Supplement. Printout of the Proofs of Theorems in Section 3.2. PDF.
• Section 3.3. The Contraction Mapping Theorem. PDF.
• Supplement. Proofs of Theorems in Section 3.3. PDF (prepared in Beamer).
• Supplement. Printout of the Proofs of Theorems in Section 3.3. PDF.
• Section 3.4. The Initial Value Problem for One Scalar Differential Equation. PDF.
• Supplement. Proofs of Theorems in Section 3.4. PDF (prepared in Beamer).
• Supplement. Printout of the Proofs of Theorems in Section 3.4. PDF.
• Section 3.5. The Initial Value Problem for Systems of Differential Equations.
• Section 3.6. An Existence Theorem for a Boundary Value Problem.