Second Course in Elementary Differential Equations

By: PAUL WALTMAN

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Paperback | 23 March 2004

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272 Pages

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Focusing on applicable rather than applied mathematics, this versatile text is appropriate for advanced undergraduates majoring in any discipline. A thorough examination of linear systems of differential equations inaugurates the text, reviewing concepts from linear algebra and basic theory. The heart of the book develops the ideas of stability and qualitative behavior. Starting with two-dimensional linear systems, the author reviews the use of polar coordinate techniques as well as Liapunov stability and elementary ideas from dynamic systems. Existence and uniqueness theorems receive a rigorous treatment, and the text concludes with a survey of linear boundary value problems. 1986 edition. 39 figures.

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Non-Fiction Mathematics Calculus & Mathematical Analysis Differential Calculus & Equations

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Non-Fiction Mathematics Calculus & Mathematical Analysis Differential Calculus & Equations

Preface	p. ix
Systems of Linear Differential Equations	p. 1
Introduction	p. 1
Some Elementary Matrix Algebra	p. 2
The Structure of Solutions of Homogeneous Linear Systems	p. 14
Matrix Analysis and the Matrix Exponential	p. 22
The Constant Coefficient Case: Real and Distinct Eigenvalues	p. 28
The Constant Coefficient Case: Complex and Distinct Eigenvalues	p. 40
The Constant Coefficient Case: The Putzer Algorithm	p. 49
General Linear Systems	p. 61
Some Elementary Stability Considerations	p. 69
Periodic Coefficients	p. 77
Scalar Equations	p. 82
An Application: Coupled Oscillators	p. 87
Two-Dimensional Autonomous Systems	p. 97
Introduction	p. 97
The Phase Plane	p. 100
Critical Points of Some Special Linear Systems	p. 108
Critical Points of General Two-Dimensional Linear Systems	p. 116
Behavior of Nonlinear Two-Dimensional Systems Near a Critical Point	p. 120
Elementary Liapunov Stability Theory	p. 132
Limit Cycles and the Poincare-Bendixon Theorem	p. 141
An Example: Lotka-Volterra Competition	p. 149
An Example: The Simple Pendulum	p. 154
Existence Theory	p. 161
Introduction	p. 161
Preliminaries	p. 162
The Contraction Mapping Theorem	p. 170
The Initial Value Problem for One Scalar Differential Equation	p. 173
The Initial Value Problem for Systems of Differential Equations	p. 181
An Existence Theorem for a Boundary Value Problem	p. 188
Boundary Value Problems	p. 195
Introduction	p. 195
Linear Boundary Value Problems	p. 196
Oscillation and Comparison Theorems	p. 206
Sturm-Liouville Problems	p. 213
The Existence of Eigenvalues for Sturm-Liouville Problems	p. 215
Two Properties of Eigenfunctions	p. 220
An Alternate Formulation - Integral Equations	p. 224
Eigenfunction Expansions	p. 225
The Inhomogeneous Sturm-Liouville Problem	p. 238
Some Standard Applications of Sturm-Liouville Theory	p. 242
Nonlinear Boundary Value Problems	p. 251
Index	p. 257
Table of Contents provided by Ingram. All Rights Reserved.

Second Course in Elementary Differential Equations

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