Book:Garrett Birkhoff/Ordinary Differential Equations/Third Edition

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Garrett Birkhoff and Gian-Carlo Rota: Ordinary Differential Equations (3rd Edition)

Published $\text {1978}$, Wiley

ISBN 0-471-05224-8


Subject Matter


Contents

PREFACE
1 FIRST-ORDER DIFFERENTIAL EQUATIONS
1 Introduction
2 Fundamental theorem of the calculus
3 First-order linear equations
4 Level curves and quasilinear DE's
5 Separable variables
6 Exact differentials; integrating factors
7 The linear fractional equation
8 Graphical integration
*9 Regular and normal curve families
10 Initial value problems
11 Uniqueness and continuity
12 The comparison theorem
2 SECOND-ORDER LINEAR EQUATIONS
1 Initial value problem
2 Constant coefficient case
3 Uniqueness theorem; Wronskian
4 Separation and comparison theorems
*5 Poincaré phase plane
6 Adjoint operators
7 Lagrange identity
8 Green's functions
9 Variation of parameters
*10 Two-endpoint problems
*11 Green's functions
3 LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS
1 The characteristic polynomial
2 Real and complex solutions
3 Linearly independent solutions
4 Solution bases
5 Stability
6 Inhomogeneous equations
7 The transfer function
*8 The Nyquist diagram
9 The Green's function
4 POWER SERIES SOLUTIONS
1 Introduction
2 Method of undetermined coefficiens
*3 Sine and cosine functions
*4 Bessel functions
5 Analytic functions
6 Method of majorants
7 First-order nonlinear differential equations
8 Undetermined coefficients
*9 Radius of convergence
*10 Method of majorants, $\text {II}$
*11 Complex solutions
5 PLANE AUTONOMOUS SYSTEMS
1 Autonomous systems
2 Plane autonomous systems
3 Poincaré phase plane
4 Linear autonomous systems
5 Equivalent systems
6 Linear equivalence
7 Stability
8 Focal, nodal, and saddle points
9 Method of Liapunov
10 Undamped nonlinear oscillations
11 Soft and hard springs
12 Damped nonlinear oscillations
*13 Limit cycles
6 EXISTENCE AND UNIQUENESS THEOREMS
1 Introduction
2 Lipschitz condition
3 Well-set problems
4 Continuity
*5 Normal systems
6 Equivalent integral equation
7 Successive approximation
8 Linear systems
9 Local existence theorem
*10 Analytic equations
11 Continuation of solutions
*12 The perturbation problem
13 Plane autonomous systems
*14 The Peano existence theorem
7 APPROXIMATE SOLUTIONS
1 Introduction
2 Cauchy polygons
3 Error bound
4 Sharper results
5 Midpoint quadrature
6 Trapezoidal quadrature
7 Trapezoidal integration
8 Order of accuracy
9 The improved Euler method
10 The modified Euler method
*11 The cumulative error
8 EFFICIENT NUMERICAL INTEGRATION
1 Introduction
2 Difference operators
3 Characteristic equation; stability
4 Polynomial interpolation
5 The interpolation error
6 Numerical differentiation
7 Roundoff errors
8 Milne's method
9 Higher-order quadrature
*10 Gaussian quadrature
11 Multistep Methods
12 Richardson Extrapolation
13 Local power series
14 Runge-Kutta method
9 REGULAR SINGULAR POINTS
1 The Continuation problem
*2 Movable singular points
3 First-order equations
4 Circuit matrix
5 Canonical bases
6 Regular singular points
7 Bessel equations
8 The fundamental theorem
*9 Alternative proof of the fundamental theorem
*10 Hypergeometric functions
*11 The Jacobi polynomials
*12 Singular points at infinity
*13 Fuchsian equations
10 STURM-LIOUVILLE SYSTEMS
1 Sturm-Liouville systems
2 Sturm-Liouville series
*3 Physical interpretations
4 Singular systems
5 Prüfer substitution
6 The Sturm comparison theorem
7 The oscillation theorem
8 The sequence of eigenfunctions
9 The Liouville normal form
10 Modified Prüfer substitution
*11 The asymptotic behavior of Bessel functions
12 Distribution of eigenvalues
13 Normalized eigenfunctions
14 Inhomogeneous equations
15 Green's functions
*16 The Schroedinger equation
*17 The square-well potential
*18 Mixed spectrum
11 EXPANSIONS IN EIGENFUNCTIONS
1 Fourier series
2 Orthogonal expansions
3 Mean-square approximations
4 Completeness
5 Orthogonal polynomials
*6 Properties of orthogonal polynomials
*7 Chebyshev polynomials
8 Euclidean vector spaces
9 Completeness of eigenfunctions
*10 Hilbert space
*11 Proof of completeness
APPENDIX A: LINEAR SYSTEMS
1 Matrix norm
2 Constant-coefficient systems
3 The matrizant
4 Floquet theorem; canonical bases
APPENDIX B: NUMERICAL INTEGRAION IN BASIC
1 Rudiments of BASIC
2 Cauchy polygon method
3 Quadrature programs
4 Improved and modified Euler methods
5 Fourth-order Runge-Kutta
BIBLIOGRAPHY
INDEX


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