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.
- Ideal for undergraduate mathematics students
- Solutions to exercises available to lecturers from solutions@cambridge.org
- Full of illustrations, worked examples, exercises and accompanying MATLAB code
Table of Contents
Introduction
Part I. First Order Differential Equations:
1. Radioactive decay and carbon dating
2. Integration variables
3. Classification of differential equations
4. Graphical representation of solutions using MATLAB
5. ‘Trivial’ differential equations
6. Existence and uniqueness of solutions
7. Scalar autonomous ODEs
8. Separable equations
9. First order linear equations and the integrating factor
10. Two ‘tricks’ for nonlinear equations
Part II. Second Order Linear Equations With Constant Coefficients:
11. Second order linear equations: general theory
12. Homogeneous 2nd order linear ODEs
13. Oscillations
14. Inhomogeneous 2nd order linear equations
15. Resonance
16. Higher order linear equations
Part III. Linear Second Order Equations With Variable Coefficients:
17. Reduction of order
18. The variation of constants formula
19. Cauchy-Euler equations
20. Series solutions of second order linear equations
Part IV. Numerical Methods and Difference Equations:
21. Euler’s method
22. Difference equations
23. Nonlinear first order difference equations
24. The logistic map
Part V. Coupled Linear Equations:
25. Vector first order equations and higher order equations
26. Explicit solutions of coupled linear systems
27. Eigenvalues and eigenvectors
28. Distinct real eigenvalues
29. Complex eigenvalues
30. A repeated real eigenvalue
31. Summary of phase portraits for linear equations
Part VI. Coupled Nonlinear Equations:
32. Coupled nonlinear equations
33. Ecological models
34. Newtonian dynamics
35. The ‘real’ pendulum
36. Periodic orbits
37. The Lorenz equations
38. What next?
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