Cover; Title Page; Copyright Page; Dedication; Preface; Contents; 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 Coefficients1. 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 Coefficients1. 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 functionsChapter 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.