Table of Contents

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.

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