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An Introduction to Ordinary Differential Equations

Book Description

This refreshing, introductory textbook covers both standard techniques for solving ordinary differential equations, as well as introducing students to qualitative methods such as phase-plane analysis. The presentation is concise, informal yet rigorous; it can be used either for 1-term or 1-semester courses. Topics such as Euler's method, difference equations, the dynamics of the logistic map, and the Lorenz equations, demonstrate the vitality of the subject, and provide pointers to further study. The author also encourages a graphical approach to the equations and their solutions, and to that end the book is profusely illustrated. The files to produce the figures using MATLAB are all provided in an accompanying website. Numerous worked examples provide motivation for and illustration of key ideas and show how to make the transition from theory to practice. Exercises are also provided to test and extend understanding: solutions for these are available for teachers.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Dedication
  6. Contents
  7. Preface
  8. Introduction
  9. Part I: First order differential equations
    1. 1. Radioactive decay and carbon dating
      1. 1.1 Radioactive decay
      2. 1.2 Radiocarbon dating
      3. Exercises
    2. 2. Integration variables
    3. 3. Classification of differential equations
      1. 3.1 Ordinary and partial differential equations
      2. 3.2 The order of a differential equation
      3. 3.3 Linear and nonlinear
      4. 3.4 Different types of solution
      5. Exercises
    4. 4. *Graphical representation of solutions using MATLAB
      1. Exercises
    5. 5. ‘Trivial’ differential equations
      1. 5.1 The Fundamental Theorem of Calculus
      2. 5.2 General solutions and initial conditions
      3. 5.3 Velocity, acceleration and Newton’s second law of motion
      4. 5.4 An equation that we cannot solve explicitly
      5. Exercises
    6. 6. Existence and uniqueness of solutions
      1. 6.1 The case for an abstract result
      2. 6.2 The existence and uniqueness theorem
      3. 6.3 Maximal interval of existence
      4. 6.4 The Clay Mathematics Institute’s $1 000 000 question
      5. Exercises
    7. 7. Scalar autonomous ODEs
      1. 7.1 The qualitative approach
      2. 7.2 Stability, instability and bifurcation
      3. 7.3 Analytic conditions for stability and instability
      4. 7.4 Structural stability and bifurcations
      5. 7.5 Some examples
      6. 7.6 The pitchfork bifurcation
      7. 7.7 Dynamical systems
      8. Exercises
    8. 8. Separable equations
      1. 8.1 The solution ‘recipe’
      2. 8.2 The linear equation x = λx
      3. 8.3 Malthus’ population model
      4. 8.4 Justifying the method
      5. 8.5 A more realistic population model
      6. 8.6 Further examples
      7. Exercises
    9. 9. First order linear equations and the integrating factor
      1. 9.1 Constant coefficients
      2. 9.2 Integrating factors
      3. 9.3 Examples
      4. 9.4 Newton’s law of cooling
      5. Exercises
    10. 10. Two ‘tricks’ for nonlinear equations
      1. 10.1 Exact equations
      2. 10.2 Substitution methods
      3. Exercises
  10. Part II: Second order linear equations with constant coefficients
    1. 11. Second order linear equations: general theory
      1. 11.1 Existence and uniqueness
      2. 11.2 Linearity
      3. 11.3 Linearly independent solutions
      4. 11.4 *The Wronskian
      5. 11.5 *Linear algebra
      6. Exercises
    2. 12. Homogeneous second order linear equations
      1. 12.1 Two distinct real roots
      2. 12.2 A repeated real root
      3. 12.3 No real roots
      4. Exercises
    3. 13. Oscillations
      1. 13.1 The spring
      2. 13.2 The simple pendulum
      3. 13.3 Damped oscillations
      4. Exercises
    4. 14. Inhomogeneous second order linear equations
      1. 14.1 Complementary function and particular integral
      2. 14.2 When f (t) is a polynomial
      3. 14.3 When f (t) is an exponential
      4. 14.4 When f (t) is a sine or cosine
      5. 14.5 Rule of thumb
      6. 14.6 More complicated functions f (t)
      7. Exercises
    5. 15. Resonance
      1. 15.1 Periodic forcing
      2. 15.2 Pseudo resonance in physical systems
      3. Exercises
    6. 16. Higher order linear equations
      1. 16.1 Complementary function and particular integral
      2. 16.2 *The general theory for nth order equations
      3. Exercises
  11. Part III: Linear second order equations with variable coefficients
    1. 17. Reduction of order
      1. Exercises
    2. 18. *The variation of constants formula
      1. Exercises
    3. 19. *Cauchy–Euler equations
      1. 19.1 Two real roots
      2. 19.2 A repeated root
      3. 19.3 Complex roots
      4. Exercises
    4. 20. *Series solutions of second order linear equations
      1. 20.1 Power series
      2. 20.2 Ordinary points
      3. 20.3 Regular singular points
      4. 20.4 Bessel’s equation
      5. Exercises
  12. Part IV: Numerical methods and difference equations
    1. 21. Euler’s method
      1. 21.1 Euler’s method
      2. 21.2 An example
      3. 21.3 *MATLAB implementation of Euler’s method
      4. 21.4 Convergence of Euler’s method
      5. Exercises
    2. 22. Difference equations
      1. 22.1 First order difference equations
      2. 22.2 Second order difference equations
      3. 22.3 The homogeneous equation
      4. 22.4 Particular solutions
      5. Exercises
    3. 23. Nonlinear first order difference equations
      1. 23.1 Fixed points and stability
      2. 23.2 Cobweb diagrams
      3. 23.3 Periodic orbits
      4. 23.4 Euler’s method for autonomous equations
      5. Exercises
    4. 24. The logistic map
      1. 24.1 Fixed points and their stability
      2. 24.2 Periodic orbits
      3. 24.3 The period-doubling cascade
      4. 24.4 The bifurcation diagram and more periodic orbits
      5. 24.5 Chaos
      6. 24.6 *Analysis of xn+1 = 4xn(1 − xn)
      7. Exercises
  13. Part V: Coupled linear equations
    1. 25. *Vector first order equations and higher order equations
      1. 25.1 Existence and uniqueness for second order equations
      2. Exercises
    2. 26. Explicit solutions of coupled linear systems
      1. Exercises
    3. 27. Eigenvalues and eigenvectors
      1. 27.1 Rewriting the equation in matrix form
      2. 27.2 Eigenvalues and eigenvectors
      3. 27.3 *Eigenvalues and eigenvectors with MATLAB
      4. Exercises
    4. 28. Distinct real eigenvalues
      1. 28.1 The explicit solution
      2. 28.2 Changing coordinates
      3. 28.3 Phase diagrams for uncoupled equations
      4. 28.4 Phase diagrams for coupled equations
      5. 28.5 Stable and unstable manifolds
      6. Exercises
    5. 29. Complex eigenvalues
      1. 29.1 The explicit solution
      2. 29.2 Changing coordinates and the phase portrait
      3. 29.3 The phase portrait for the original equation
      4. Exercises
    6. 30. A repeated real eigenvalue
      1. 30.1 A is a multiple of the identity: stars
      2. 30.2 A is not a multiple of the identity: improper nodes
      3. Exercises
    7. 31. Summary of phase portraits for linear equations
      1. 31.1 *Jordan canonical form
      2. Exercises
  14. Part VI: Coupled nonlinear equations
    1. 32. Coupled nonlinear equations
      1. 32.1 Some comments on phase portraits
      2. 32.2 Competition of species
      3. 32.3 Direction fields
      4. 32.4 Analytical method for phase portraits
      5. Exercises
    2. 33. Ecological models
      1. 33.1 Competing species
      2. 33.2 Predator-prey models I
      3. 33.3 Predator-prey models II
      4. Exercises
    3. 34. Newtonian dynamics
      1. 34.1 One-dimensional conservative systems
      2. 34.2 *A bead on a wire
      3. 34.3 Dissipative systems
      4. Exercises
    4. 35. The ‘real’ pendulum
      1. 35.1 The undamped pendulum
      2. 35.2 The damped pendulum
      3. 35.3 Alternative phase space
      4. Exercises
    5. 36. *Periodic orbits
      1. 36.1 Dulac’s criterion
      2. 36.2 The Poinacré–Bendixson Theorem
      3. Exercises
    6. 37. *The Lorenz equations
    7. 38. What next?
      1. 38.1 Partial differential equations and boundary value problems
      2. 38.2 Dynamical systems and chaos
      3. Exercises
  15. Appendix A: Real and complex numbers
  16. Appendix B: Matrices, eigenvalues, and eigenvectors
  17. Appendix C: Derivatives and partial derivatives
  18. Index