Synopses & Reviews
This concise treatment of differential equations is intended to serve as a text for a standard one-semester or two-term undergraduate course in differential equations following the calculus. Emphasis is placed on mathematical explanations-ranging from routine calculations to moderately sophisticated theorems-in order to import more than a rote understanding of techniques.Beginning with a survey of first order equations, the text goes on to consider linear equations — including discussions of complex-valued solutions, linear differential operators, inverse operators, and variations of parameters method. Subsequent chapters then examine the Laplace transform and Picards existence theorem and conclude with an exploration of various interpretations of systems of equations.
Synopsis
Concise introduction for undergraduates includes, among other topics, a survey of first order equations, discussions of complex-valued solutions, linear differential operators, inverse operators and variation of parameters method, the Laplace transform, Picard's existence theorem, and an exploration of various interpretations of systems of equations. Numerous clearly stated theorems and proofs, examples, and problems followed by solutions.
Synopsis
First-rate introduction for undergraduates examines first order equations, complex-valued solutions, linear differential operators, the Laplace transform, Picard's existence theorem, and much more. Includes problems and solutions.
Table of Contents
Chapter 1. First Order Equations
1.1 Introduction
1.2 Variable separate
1.3 Geometric interpretation of first order equations
1.4 Existence and uniqueness theorem
1.5 Families of curves and envelopes
1.6 Clairaut equations
Chapter 2. Special Methods for First Order Equations
2.1 Homogeneous equations--substitutions
2.2 Exact equations
2.3 Line integrals
2.4 First order linear equations--integrating factors
2.5 Orthogonal families
2.6 Review of power series
2.7 Series solutions
Chapter 3. Linear Equations
3.1 Introduction
3.2 Two theorems on linear algebraic equations
3.3 General theory of linear equations
3.4 Second order equations with constant coefficients
3.5 Applications
Chapter 4. Special Methods for Linear Equations
4.1 Complex-valued solutions
4.2 Linear differential operators
4.3 Homogeneous equations with constant coefficients
4.4 Method of undetermined coefficients
4.5 Inverse operators
4.6 Variation of parameters method
Chapter 5. The Laplace Transform
5.1 Review of improper integrals
5.2 The Laplace transform
5.3 Properties of the transform
5.4 Solution of equations by transforms
Chapter 6. Picard's Existence Theorem
6.1 Review
6.2 Outline of the Picard method
6.3 Proof of existence and uniqueness
6.4 Approximations to solutions
Chapter 7. Systems of Equations
7.1 Geometric interpretation of a system
7.2 Other interpretations of a system
7.3 A system equivalent to M(x,y)dx+N(x,y)dy=0
7.4 Existence and uniqueness theorems
7.5 Existence theorem for nth order equations
7.6 Polygonal approximations for systems
7.7 Linear systems
7.8 Operator methods
7.9 Laplace transform methods
Index