TY  - BOOK
AU  - Birkhoff, Garrett.
AU  - Rota, Gian-Carlo.
TI  - Ordinary differential equations
SN  - 9812530029
U1  - 515.35 
PY  - 2003///
CY  - New York
PB  - John Wiley & Sons
KW  - Differential equations
N1  - 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
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