ORDINARY DIFFERENTIAL EQUATION 
FOURTH EDITION
Garrett Birkhoff, Harvard
Gian-Carlo Rota, MIT

JOHN WILEY & SONS 
New York, Chichester,  Brisbane, Toronto, Singapore

1 FIRST-ORDER OF DIFFERENTIAL EQUATIONS 1

1. Introduction 1
2. Fundamental Theorem of the Calculus 2
3. First-order Linear Equations 7
4. Separable Equations 9
5. Quasilinear Equations; Implicit Solutions 11 
6. Exact Differentials; Integrating Factors 15 
7. Linear Fractional Equations 17
8. Graphical and Numerical Integration 20
9. The Initial Value Problem 24
10. Uniqueness and Continuity 26 
11. A Comparison Theorem 29
12. Regular and Normal Curve Families 31

2 SECOND-ORDER LINEAR EQUATIONS 34

1. Bases of Solutions 34
2. Initial Value Problems 37
3. Qualitative Behavior; Stability 39
4. Uniqueness Theorem 40
5. The Wronskian 43
6. Separation and Comparison Theorems 47 
7. The Phase Plane 49
8. Adjoint Operators; Lagrange Identity 54 
9. Green's Functions 58
10. Two-endpoint Problems 63
11. Green's Functions, II 65

3 LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS 71

1. The Characteristic Polynomial 71
2. Complex Exponential Functions 72 
3. The Operational Calculus 76
4. Solution Bases 78
5. Inhomogeneous Equations 83
6. Stability 85
7. The Transfer Function 86
8. The Nyquist Diagram 90 
9. The Green's Function 93

4 POWER SERIES SOLUTIONS 99

1. Introduction 99
2. Method of Undetermined Coefficients 101 
3. More Examples 105
4. Three First-order DEs 107
5. Analytic Functions 110
6. Method of Majorants 113
7. Sine and Cosine Functions 116
8. Bessel Functions 117
9. First-order Nonlinear DEs 121
10. Radius of Convergence 124 
ll. Method of Majorants, II 126 
12. Complex Solutions 128


5 PLANE AUTONOMOUS SYSTEMS 131

1. Autonomous Systems 131
2. Plane Autonomous Systems 134
3. The Phase Plane, II 136
4. Linear Autonomous Systems 141
5. Linear Equivalence 144
6. Equivalence Under Diffeomorphisms 151 
7. Stability 153
8. Method of Liapunov 157
9. Undamped Nonlinear Oscillations 158
10. Soft and Hard Springs 159
11. Damped Nonlinear Oscillations 163
12. Limit Cycles 164


6 EXISTENCE AND UNIQUENESS THEOREMS 170

1. Introduction 170
2. Lipschitz conditions 172
3. Well-posed Problems 174
4. Continuity 177
5. Normal Systems 180
6. Equivalent Integral Equation 183
7. Successive Approximation 185 
8. Linear Systems 188
9. Local Existence Theorem 190
10. The Peano Existence Theorem 191 
ll. Analytic Equations 193
12. Continuation of Solutions 197 
13. The Perturbation Equation 198

7 APPROXIMATE SOLUTIONS 204

1. Intoduction 204
2. Error Bounds 205
3. Deviation and Error 207
4. Mesh-halving; Richardson Extrapolation 210 
5. Midpoint Quadrature 212
6. Trapezoidal Quadrature 215
7. Trapezoidal Integration 218
8. The Improved Euler Method 222
9. The Modified Euler Method 224
10. Cumulative Error Bound 226

8 EFFICIENT NUMERICAL INTEGRATION 230

1. Difference Operators 230
2. Polynomial Interpolation 232
3. Interpolation Errors 235
4. Stability 237
5. Numerical Differentiation; Roundoff 240 
6. Higher Order Quadrature 244
7. Gaussian Quadrature 248
8. Fourth-order Runge-Kutta 250
9. Milne's Method 256
10. Multistep Methods 258

9 REGULAR SINGULAR POINTS 261

1. Introduction 261
2. Movable Singular Points 263
3. First-order Linear Equations 264
4. Continuation Principle; Circuit Matrix 268
5. Canonical Bases 270
6. Regular Singular Points 274
7. Bessel Equation 276
8. The Fundamental Theorem 281
9. Alternative Proof of the Fundamental Theorem 285 
10. Hypergeometric Functions 287
ll. The Jacobi Polynomials 289
12. Singular Points at Infinity 292
13. Fuchsian Equations 294

10 STURM-LIOUVILLE SYSTEMS 300

1. Sturm-Liouville Systems 300
2. Sturm-Liouville Series 302
3. Physical Interpretations 305
4. Singular Systems 308
5. Priifer Substitution 312
6. Sturm Comparison Theorem 313
7. Sturm Oscillation Theorem 314
8. The Sequence of Eigenfunctions 318
9. The Liouville Normal Form 320
10. Modified Priifer Substitution 323
11. The Asymptotic Behavior of Bessel Functions 326 
12. Distribution of Eigenvalues 328
13. Normalized Eigenfunctions 329
14. Inhomogeneous Equations 333
15. Green's Functions 334
16. The Schroedinger Equation 336
17. The Square-well Potential 338
18. Mixed Spectrum 339

11 EXPANSIONS IN EIGENFUNCTIONS 344

1. Fourier Series 344
2. Orthogonal Expansions 346
3. Mean-square Approximation 347
4. Completeness 350
5. Orthogonal Polynomials 352
6. Properties of Orthogonal Polynomials 354
7. Chebyshev Polynomials 358
8. Euclidean Vector Spaces 360
9. Completeness of Eigenfunctions 363
10. Hilbert Space 365
11. Proof of Completeness 367

APPENDIX A: LINEAR SYSTEMS 371

1. Matrix Norm 371
2. Constant-coefficient Systems 372
3. The Matrizant 375
4. Floquet Theorem; Canonical Bases 377


APPENDIX B: BIFURCATION THEORY 380

1. What Is Bifurcation? 380
2. Poincare Index Theorem 381
3. Hamiltonian Systems 383
4. Hamiltonian Bifurcations 386 
5. Poincare Maps 387
6. Periodically Forced Systems 389