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## 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.

1. Cover
2. Half Title
3. Title Page
5. Dedication
6. Contents
7. Preface
8. Introduction
9. Part I: First order differential equations
1. 1. Radioactive decay and carbon dating
2. 2. Integration variables
3. 3. Classification of differential equations
4. 4. *Graphical representation of solutions using MATLAB
5. 5. ‘Trivial’ differential equations
6. 6. Existence and uniqueness of solutions
7. 7. Scalar autonomous ODEs
8. 8. Separable equations
9. 9. First order linear equations and the integrating factor
10. 10. Two ‘tricks’ for nonlinear equations
10. Part II: Second order linear equations with constant coefficients
1. 11. Second order linear equations: general theory
2. 12. Homogeneous second order linear equations
3. 13. Oscillations
4. 14. Inhomogeneous second order linear equations
5. 15. Resonance
6. 16. Higher order linear equations
11. Part III: Linear second order equations with variable coefficients
1. 17. Reduction of order
2. 18. *The variation of constants formula
3. 19. *Cauchy–Euler equations
4. 20. *Series solutions of second order linear equations
12. Part IV: Numerical methods and difference equations
1. 21. Euler’s method
2. 22. Difference equations
3. 23. Nonlinear first order difference equations
4. 24. The logistic map
13. Part V: Coupled linear equations
1. 25. *Vector first order equations and higher order equations
2. 26. Explicit solutions of coupled linear systems
3. 27. Eigenvalues and eigenvectors
4. 28. Distinct real eigenvalues
5. 29. Complex eigenvalues
6. 30. A repeated real eigenvalue
7. 31. Summary of phase portraits for linear equations
14. Part VI: Coupled nonlinear equations
1. 32. Coupled nonlinear equations
2. 33. Ecological models
3. 34. Newtonian dynamics
4. 35. The ‘real’ pendulum
5. 36. *Periodic orbits
6. 37. *The Lorenz equations
7. 38. What next?
15. Appendix A: Real and complex numbers
16. Appendix B: Matrices, eigenvalues, and eigenvectors
17. Appendix C: Derivatives and partial derivatives
18. Index