Introduction to Ordinary Differential Equations

Chapter 0. Preliminaries 1. Introduction 2 Complex numbers 3 Functions 4 Polynomials 5. Complex series and the exponential function 6. Determinants 7. Remarks on methods of discovery and proof Chapter 1. Introduction--Linear Equations of the First Order 1. Introduction 2. Differential equations 3. Problems associated with differential equations 4. Linear equations of the first order 5. The equation y'+ay=0 6. The equation y'+ay=b(x) 7. The general linear equation of the first order Chapter 2. Linear Equations with Constant Coefficients 1. Introduction 2. The second order homogeneous equation 3. Initial value problems for second order equations 4. Linear dependence and independence 5. A formula for the Wronskian 6. The non-homogeneous equation of order two 7. The homogeneous equation of order n 8. Initial value problems for n-th order equations 9. Equations with real constants 10. The non-homogeneous equation of order n 11. A special method for solving the non-homogeneous equation 12. Algebra of constant coefficient operators Chapter 3. Linear Equations with Variable Coefficients 1. Introduction 2. Initial value problems for the homogeneous equation 3. Solutions of the homogeneous equation 4. The Wronskian and linear independence 5. Reduction of the order of a homogeneous equation 6. The non-homogeneous equation 7. Homogeneous equations with analytic coefficients 8. The Legendre equation 9. Justification of the power series method Chapter 4. Linear Equations with Regular Singular Points 1. Introduction 2. The Euler equation 3. Second order equations with regular singular points--an example 4. Second order equations with regular singular points--the general case 5. A convergence proof 6. The exceptional cases 7. The Bessel equation 8. The Bessel equation (continued) 9. Regular singular points at infinity Chapter 5. Existence and Uniqueness of Solutions to First Order Equations 1. Introduction 2. Equations with variables separated 3. Exact equations 4. The method of successive approximations 5. The Lipschitz condition 6. Convergence of the successive approximations 7. Non-local existence of solutions 8. Approximations to, and uniqueness of, solutions 9. Equations with complex-valued functions Chapter 6. Existence and Uniqueness of Solutions to Systems and n-th Order Equations 1. Introduction 2. An example--central forces and planetary motion 3. Some special equations 4. Complex n-dimensional space 5. Systems as vector equations 6. Existence and uniqueness of solutions to systems 7. Existence and uniqueness for linear systems 8. Equations of order n References; Answers to Exercises; Index