# Ordinary Differential Equations: A Practical Guide

###### Bernd J. Schroers
Publisher:
Cambridge University Press
Publication Date:
2011
Number of Pages:
118
Format:
Paperback
Series:
AIMS Library Series
Price:
25.99
ISBN:
9781107697492
Category:
Textbook
[Reviewed by
Allen Stenger
, on
06/29/2012
]

This is a chatty and easy-to-read brief introductory cookbook of ordinary differential equations. The extent of coverage is roughly what is found in good engineering math books (for example, Kreyszig’s Advanced Engineering Mathematics). The most conspicuous omission is the Laplace transform. There are also no numerical methods, although the reader is encouraged to use his favorite computer algebra system to investigate some examples numerically.

The book puts emphasis on qualitative methods, with an especially lengthy discussion of phase diagrams. There are exercises scattered throughout, with various degrees of difficulty, and a variety of applications are given (with the models assumed rather than developed). The book concludes with five term projects for groups, that require some modeling.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.

1. Preface
1. First order differential equations
• 1.1 General remarks about differential equations
• 1.1.1 Terminology
• 1.1.2 Approaches to problems involving differential equations
• 1.2 Exactly solvable first order ODEs
• 1.2.1 Terminology
• 1.2.2 Solution by integration
• 1.2.3 Separable equations
• 1.2.4 Linear first order differential equations
• 1.2.5 Exact equations
• 1.2.6 Changing variables
• 1.3 Existence and uniqueness of solutions
• 1.4 Geometric methods: direction fields
• 1.5 Remarks on numerical methods
2. Systems and higher order equations
• 2.1 General remarks
• 2.2 Existence and uniqueness of solutions for systems
• 2.3 Linear systems
• 2.3.1 General remarks
• 2.3.2 Linear algebra revisited
• 2.4 Homogeneous linear systems
• 2.4.1 The vector space of solutions
• 2.4.2 The eigenvector method
• 2.5 Inhomogeneous linear systems
3. Second order equations and oscillations
• 3.1 Second order differential equations
• 3.1.1 Linear, homogeneous ODEs with constant coefficients
• 3.1.2 Inhomogeneous linear equations
• 3.1.3 Euler equations
• 3.1.4 Reduction of order
• 3.2 The oscillating spring
• 3.2.1 Deriving the equation of motion
• 3.2.2 Unforced motion with damping
• 3.2.3 Forced motion with damping
• 3.2.4 Forced motion without damping
4. Geometric methods
• 4.1 Phase diagrams
• 4.1.1 Motivation
• 4.1.2 Definitions and examples
• 4.1.3 Phase diagrams for linear systems
• 4.2 Nonlinear systems
• 4.2.1 The Linearisation Theorem
• 4.2.2 Lyapunov functions
5. Projects
• 5.1 Ants on polygons
• 5.2 A boundary value problem in mathematical physics
• 5.3 What was the trouble with the Millennium Bridge?
• 5.4 A system of ODEs arising in differential geometry
• 5.5 Proving the Picard-Lindelöf Theorem
• 5.5.1 The Contraction Mapping Theorem
• 5.5.2 Strategy of the proof
• 5.5.3 Completing the proof
1. References
2. Index