An Introduction To Ordinary Differential Equations

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This book is designed as a text for a first course in ordinary differential equations. The author has given an elementary, thorough, systematic introduction tothe subject. The emphasis throughout is on general properties of equations and their solutions. There are many exercises which help the students to develop the techniques of solving equations. The problems in the text are intended to help sharpen the student’ sunder standing of the subject. The author has use dexercises to introduce the students to various new topics such as stability equations with periodic coefficients and boundary value problems. The only prerequisite to understand this text is the knowledge of calculus. Table Of Contents PRELIMINARIES Introduction. Complex Numbers. Functions. Polynomials. Complex Series and the Exponential Function. Determinants. Remarks on Methods of Discovery and Proof. INTRODUCTION LINEAR EQUATIONS OF THE FIRST ORDER Introduction. Differential Equations. Problems Associated with Differential Equations. Linear Equations of the First Order. The Equation y´ + ay =0. The Equation y´ + ay = b(x). The General Linear Equation of the First Order. LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS Introduction. The Second Order Homogeneous Equation. Initial Value Problems for Second Order Equations. Linear Dependence and Independence. A Formula for the Wronskian. The Non-Homogeneous Equation of Order Two. The Homogeneous Equation of Order n. Initial Value Problems for n-th Order Equations. Equations with Real Constants. The Non-Homogeneous Equation of Order n. A Special Method for Solving the Non-Homogeneous Equation. Algebra of Constant Coefficient Operators. LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS Introduction. Initial Value Problems for the Homogeneous Equation. Solutions of the Homogeneous Equation. The Wronskian and Linear Independence. Reduction of the Order of a Homogeneous Equation. The Non-Homogeneous Equation. Homogeneous Equations with Analytic Coefficients. The Legendre Equation. Justification of the Power Series Method. LINEAR EQUATIONS WITH REGULAR SINGULAR POINTS Introduction. The Euler Equation. Second Order Equations with Regular Singular Points—an Example. Second Order Equations with Regular Singular Points—the General Case. A Convergence Proof. The Exceptional Cases. The Bessel Equation. Regular Singular Points at Infinity. EXISTENCE AND UNIQUENESS OF SOLUTIONS TO FIRST ORDER EQUATIONS Introduction. Equations with Variables Separated. Exact Equations. The Method of Successive Approximations. The Lipschitz Condition. Convergence of the Successive Approximations. Non-local Existence of Solutions. Approximations to, and Uniqueness of, Solutions. Equations with Complex-valued Functions. EXISTENCE AND UNIQUENESS OF SOLUTIONS TO SYSTEMS AND ORDER EQUATIONS Introduction. An Example—Central Forces and Planetary Motion. Some Special Equations. Complex n-dimensional Space. Systems as Vector Equations. Existence and Uniqueness of Solutions to Systems. Existence and Uniqueness for Linear Systems. Equations of Order n. References. Answers to Exercises. Index.