1 Solutions of ODEs and Their Properties. - 1. 1 Introduction. - 1. 2 Definitions and Existence Theory. - 1. 3 Functions DSolve NDSolve and Differentiallnvariants. - 1. 4 The Phase Portrait. - 1. 5 Applications of the Programs Sysn Phase2D PolarPhase and Phase3D. - 1. 6 Problems. - 2 Linear ODEs with Constant Coefficients. - 2. 1 Introduction. - 2. 2 The General Solution of Linear Differential Systems with Constant Coefficients. - 2. 3 The Program LinSys. - 2. 4 Problems. - 3 Power Series Solutions of ODEs and Frobenius Series. - 3. 1 Introduction. - 3. 2 Power Series and the Program Taylor. - 3. 3 Power Series and Solutions of ODEs. - 3. 4 Series Solutions Near Regular Singular Points: Method of Frobenius. - 3. 5 The Program SerSol. - 3. 6 Other Applications of SerSol. - 3. 7 The Program Frobenius. - 3. 8 Problems. - 4 Poincaré's Perturbation Method. - 4. 1 Introduction. - 4. 2 Poincaré's Perturbation Method. - 4. 3 How to Introduce the Small Parameter. - 4. 4 The Program Poincare. - 4. 5 Problems. - 5 Problems of Stability. - 5. 1 Introduction. - 5. 2 Definitions of Stability. - 5. 3 Analysis of Stability: The Direct Method. - 5. 4 Polynomial Liapunov Functions. - 5. 5 The Program Liapunov. - 5. 6 Analysis of Stability the Indirect Method: The Planar Case. - 5. 7 The Program LStability. - 5. 8 Problems. - 6 Stability: The Critical Case. - 6. 1 Introduction. - 6. 2 The Planar Case and Poincaré's Method. - 6. 3 The Programs CriticalEqS and CriticalEqN. - 6. 4 The Center Manifold. - 6. 5 The Program CManifold. - 6. 6 Problems. - 7 Bifurcation in ODEs. - 7. 1 Introduction to Bifurcation. - 7. 2 Bifurcation in a Differential Equation Containing One Parameter. - 7. 3 The Programs Bifl and Bif1G. - 7. 4 Problems. - 7. 5 Bifurcation in a Differential Equation Depending on Two Parameters. - 7. 6 The Programs Bif2 and Bif2G. - 7. 7 Problems. - 7. 8 Hopf'sBifurcation. - 7. 9 The Program HopfBif. - 7. 10 Problems. - 8 The Lindstedt-Poincaré Method. - 8. 1 Asymptotic Expansions. - 8. 2 The Lindstedt-Poincaré Method. - 8. 3 The Programs LindPoinc and GLindPoinc. - 8. 4 Problems. - 9 Boundary-Value Problems for Second-Order ODEs. - 9. 1 Boundary-Value Problems and Bernstein's Theorem. - 9. 2 The Shooting Method. - 9. 3 The Program NBoundary. - 9. 4 The Finite Difference Method. - 9. 5 The Programs NBoundaryl and NBoundary2. - 9. 6 Problems. - 10 Rigid Body with a Fixed Point. - 10. 1 Introduction. - 10. 2 Euler's Equations. - 10. 3 Free Rotations or Poinsot's Motions. - 10. 4 Heavy Gyroscope. - 10. 5 The Gyroscopic Effect. - 10. 6 The Program Poinsot. - 10. 7 The Program Solid. - 10. 8 Problems. - A How to Use the Package ODE. m. - References. Language: English