# A Second Course in Elementary Differential Equations

## Description

A Second Course in Elementary Differential Equations deals with norms, metric spaces, completeness, inner products, and an asymptotic behavior in a natural setting for solving problems in differential equations. The book reviews linear algebra, constant coefficient case, repeated eigenvalues, and the employment of the Putzer algorithm for nondiagonalizable coefficient matrix. The text describes, in geometrical and in an intuitive approach, Liapunov stability, qualitative behavior, the phase plane concepts, polar coordinate techniques, limit cycles, the Poincaré-Bendixson theorem. The book explores, in an analytical procedure, the existence and uniqueness theorems, metric spaces, operators, contraction mapping theorem, and initial value problems. The contraction mapping theorem concerns operators that map a given metric space into itself, in which, where an element of the metric space M, an operator merely associates with it a unique element of M. The text also tackles inner products, orthogonality, bifurcation, as well as linear boundary value problems, (particularly the Sturm-Liouville problem). The book is intended for mathematics or physics students engaged in ordinary differential equations, and for biologists, engineers, economists, or chemists who need to master the prerequisites for a graduate course in mathematics.

## Table of Contents

Preface
1 Systems of Linear Differential Equations
1. Introduction
2. Some Elementary Matrix Algebra
3. The Structure of Solutions of Homogeneous Linear Systems
4. Matrix Analysis and the Matrix Exponential
5. The Constant Coefficient Case: Real and Distinct Eigenvalues
6. The Constant Coefficient Case: Complex and Distinct Eigenvalues
7. The Constant Coefficient Case: The Putzer Algorithm
8. General Linear Systems
9. Some Elementary Stability Considerations
10. Periodic Coefficients
11. Scalar Equations
12. An Application: Coupled Oscillators
2 Two-Dimensional Autonomous Systems
1. Introduction
2. The Phase Plane
3. Critical Points of Some Special Linear Systems
4. Critical Points of General TWo-Dimensional Linear Systems
5. Behavior of Nonlinear TWo-Dimensional Systems Near a Critical Point
6. Elementary Liapunov Stability Theory
7. Limit Cycles and the Poincaré-Bendixson Theorem
8. An Example: Lotka-Volterra Competition
9. An Example: The Simple Pendulum
3 Existence Theory
1. Introduction
2. Preliminaries
3. The Contraction Mapping Theorem
4. The Initial Value Problem for One Scalar Differential Equation
5. The Initial Value Problem for Systems of Differential Equations
6. An Existence Theorem for a Boundary Value Problem
4 Boundary Value Problems
1. Introduction
2. Linear Boundary Value Problems
3. Oscillation and Comparison Theorems
4. Sturm-Liouville Problems
5. The Existence of Eigenvalues for Sturm-Liouville Problems
6. Twο Properties of Eigenfunctions
7. An Alternate Formulation-Integral Equations
8. Eigenfunction Expansions
9. The Inhomogeneous Sturm-Liouville Problem
10. Some Standard Applications of Sturm-Liouville Theory
11. Nonlinear Boundary Value Problems
Index