Differential Equations: A Concise Course

CHAPTER 1  FIRST ORDER EQUATIONS
     1-1 Introduction
     1-2 Variables 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

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