# Description

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

This refreshing, introductory textbook covers both standard techniques for solving ordinary , as well as introducing to qualitative methods such as phase-plane . 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 to the equations and their solutions, and to that end the 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.

# Table of Content

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?