Ordinary Differential Equations

1. INTRODUCTION AND FIRST-ORDER EQUATIONS. What is an Ordinary Differential Equation?. Linear Equations. Other Solvable Equations. Inequalities, Uniqueness, and Existence. 2. THE GENERAL LINEAR EQUATION. Second-order Linear Equations. The Equation of Order n. The First-order System. 3. LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS. Complex Numbers and Polynomials. The Exponential Function. Bases of Solutions. The Inhomogeneous Equation. The System Formulation. 4. ANALYTIC EQUATIONS. Power Series. Linear, Analytic Equations. Power-series Expansions for n = 2. The Method of Majorants. Higher-Order Equations and Systems. 5. ISOLATED SINGULAR POINTS. The Euler Equation. The Method of Frobenius. Convergence. 6. EXISTENCE AND REGULARITY OF SOLUTIONS. The First-order System. The System of n First-order Equations. Continuity of Solutions. Differentiability. 7. DYNAMICAL SYSTEMS. Autonomous Equations. Constant and Periodic Solutions. Limit Sets. Poincare-Bendixson Theory. 8. STABILITY. Linear Stability. Nonlinear Stability of Equilibrium Points. Stability of Periodic Solutions. 9. BIFURCATION THEORY. Low-dimensional Maps and Their Bifurcations. Bifurcations of Solutions of Differential Equations. Chaotic Dynamics of Maps. Melnikov's Integral and Chaotic Dynamics. 10. BOUNDARY-VALUE PROBLEMS. Operators and Their Adjoints. Eigenvalue Problems. The Sturm-Liouville Problem. Green's Functions. 11. STURM-LIOUVILLE THEORY. The Prufer Substitution. Comparison Theorems. The Sturm-Liouville Theorem. 12. EIGENFUNCTION EXPANSIONS. Kinds of Convergence. Inner Product and Orthogonality. Extremal Properties of Eigenvalues. The Norm of an Operator. The Sequence of Eigenfunctions. Relation of Assumptions.