Book:Garrett Birkhoff/Ordinary Differential Equations/Third Edition

Subject Matter

 * Ordinary Differential Equations

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



Source Work Progress
* : Chapter $1$ First-Order Differential Equations: $2$ Fundamental Theorem of the Calculus