Ordinary Differential Equations A Practical Guide

• Main Author: Schroers, Bernd J.
• Format: eBook
• Language: English
• Published: Cambridge Cambridge University Press 29.09.2011
• Edition: 1
• Series: African Institute of Mathematics library series
• Subjects: Differential equations

Ordinary Differential Equations introduces key concepts and techniques in the field and shows how they are used in current mathematical research and modelling. It deals specifically with initial value problems, which play a fundamental role in a wide range of scientific disciplines, including mathem...

Table of Contents:

• Cover -- ORDINARY DIFFERENTIAL EQUATIONS -- African Institute of Mathematics Library Series -- Title -- Copyright -- Contents -- 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 -- References -- Index