Ordinary differential equations/ Garrett Birkhoff, Gian-Carlo Rota.

1 FIRST-ORDER OF DIFFERENTIAL EQUATIONS 1
1. Introduction
2. Fundamental Theorem of the Calculus
3. First-order Linear Equations
4. Separable Equations
5. Quasilinear Equations; Implicit Solutions
6. Exact Differentials; Integrating Factors
7. Linear Fractional Equations
8. Graphical and Numerical Integration
9. The Initial Value Problem
10. Uniqueness and Continuity
11. A Comparison Theorem
12. Regular and Normal Curve Families
2 SECOND-ORDER LINEAR EQUATIONS
1. Bases of Solutions
2. Initial Value Problems
3. Qualitative Behavior; Stability
4. Uniqueness Theorem
5. The Wronskian
6. Separation and Comparison Theorems
7. The Phase Plane
8. Adjoint Operators; Lagrange Identity
9. Green's Functions
10. Two-endpoint Problems
11. Green's Functions,
3 LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS
1. The Characteristic Polynomial
2. Complex Exponential Functions
3. The Operational Calculus
4. Solution Bases
5. Inhomogeneous Equations
Contents
6. Stability
7. The Transfer Function
8. The Nyquist Diagram
9. The Greenes Function
4 POWER SERIES SOLUTIONS
1. Introduction
2. Method of Undetermined Coefficients
3. More Examples
4. Three First-order DEs
5. Analytic Functions
6. Method of Majorants
*7. Sine and Cosine Functions
*8. Bessel Functions
9. First-order Nonlinear DEs
10. Radius of Convergence
11. Method of Majorants,
12. Complex Solutions
5 PLANE AUTONOMOUS SYSTEMS
1. Autonomous Systems
2. Plane Autonomous Systems
3. The Phase Plane,
4. Linear Autonomous Systems
5. Linear Equivalence
6. Equivalence Under Diffeomorphisms
7. Stability
8. Method of Liapunov
9. Undamped Nonlinear Oscillations
10. Soft and Hard Springs
11. Damped Nonlinear Oscillations
12. Limit Cycles
6 EXISTENCE AND UNIQUENESS THEOREMS
1. Introduction
2. Lipschitz conditions
3. Well-posed Problems
4. Continuity
*5. Normal Systems
6. Equivalent Integral Equation
7. Successive Approximation
8. Linear Systems
9. Local Existence Theorem
Contents
10. The Peano Existence Theorem
11. Analytic Equations
12. Continuation of Solutions
13. The Perturbation Equation
7 APPROXIMATE SOLUTIONS
1. Introduction
2. Error Bounds
*3. Deviation and Error
4. Mesh-halving; Richardson Extrapolation
5. Midpoint Quadrature
6. Trapezoidal Quadrature
7. Trapezoidal Integration
8. The Improved Euler Method
9. The Modified Euler Method
10. Cumulative Error Bound 226
8 EFFICIENT NUMERICAL INTEGRATION
1. Difference Operators
2. Polynomial Interpolation
3. Interpolation Errors
4. Stability
5. Numerical Differentiation; Roundoff
6. Higher Order Quadrature
7. Gaussian Quadrature
8. Fourth-order Runge-Kutta
9. Millie's Method
10. Multistep Methods
9 REGULAR SINGULAR POINTS
1. Introduction
2. Movable Singular Points
3. First-order Linear Equations
4. Continuation Principle; Circuit Matrix
5. Canonical Bases
6. RegularSingular Points
7. Bessel Equation
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-Liouvilie Systems
2. Sturm-Liouville Series
3. Physical Interpretations
4. Singular Systems
5. Prufer Substitution
6. Sturm Comparison Theorem
7. Sturm Oscillation Theorem
8. The Sequence of Eigenfunctions
9. The Liouville Normal Form
10. Modified Priifer 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 Approximation
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: BIFURCATION THEORY
1. What Is Bifurcation?
2. Poincare Index Theorem
3. Hamiltonian Systems
4. Hamiltonian Bifurcations
5. Poincare Maps
6. Periodically Forced Systems
BIBLIOGRAPHY
INDEX