An Introduction to Ordinary Differential Equations

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ISBN: 9780486659428 | 0486659429
Cover: Paperback
Copyright: 3/1/1989

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A thorough, systematic first course in elementary differential equations for undergraduates in mathematics and science, requiring only basic calculus for a background, and including many exercises designed to develop students' technique in solving equations. With problems and answers. Index.

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
The Bessel equation (continued)
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 n-th 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
Table of Contents provided by Publisher. All Rights Reserved.

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An Introduction to Ordinary Differential Equations

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