This introduction to ordinary differential and difference equations is suited not only for mathematicians but for scientists and engineers as well. Exact solutions methods and qualitative approaches are covered, and many illustrative examples are included.

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.

Matlab is used to generate graphical representations of solutions. The files to produce the figures using MATLAB are all provided in an accompanying file. Numerous worked examples provide motivation for and illustration of key ideas and show how to make the transition from theory to practice.

**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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