Integration and differential equations

Description

Integration involves ideas, with associated techniques, that are familiar from school mathematics; mastering this branch of mathematics is an essential requirement before moving to more sophisticated concepts and applications. The material in this text (Part I) introduces and develops the standard techniques of elementary integration and, in some cases, takes the ideas a little further. In Part II, the concept of an ordinary differential equation is explored, and the solution-methods for most of the standard types are explained and developed. Many worked examples are included.

Contenu

1. Introduction and Background
   1. The Riemann integral
   2. The fundamental theorem of calculus
   3. Theorems on integration
   4. Standard integrals
   5. Integration by parts
   6. Improper integrals
   7. Non-uniqueness of representation
   8. Exercises 1
2. The integration of rational functions
   1. Improper fractions
   2. Linear denominator, Q(x)
   3. Quadratic denominator, Q(x)
   4. Cubic denominator, Q(x)
   5. Exercises 2
3. The integration of trigonometric functions
   1. Simple products
   2. Powers of sin and cos
   3. Rational functions of sin and cos
   4. Exercises 3
   5. Answers
4. Part II The integration of ordinary differential equations
5. List of Equations
6. Preface
7. What is a differential equation ?
   1. The nature and solution of differential equations
   2. Classification of ODEs
   3. Overview of the equations to be discussed
8. First order ODEs: standard results
   1. Separable equations
   2. Homogeneous equations
   3. The general linear equation
9. First order ODEs: special equations
   1. The Bernoulli equation
   2. The Clairaut equation
   3. The Riccati equation
   4. Exact differentials
   5. Missing variables
10. Second order ODEs
    1. Constant coefficient equations
    2. The Euler equation
    3. Reduction of order
    4. Variation of parameters
    5. Finding particular integrals directly
    6. Missing variables
    7. Exercises 4
11. More general aspects of ODEs
    1. Interchanging x and y
    2. Singular solutions
    3. Uniqueness
    4. Exercises 5
    5. Answers
    6. Index