Book contents
- Frontmatter
- Contents
- Preface
- Introduction
- Part I First order differential equations
- Part II Second order linear equations with constant coefficients
- Part III Linear second order equations with variable coefficients
- Part IV Numerical methods and difference equations
- Part V Coupled 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?
- Appendix A Real and complex numbers
- Appendix B Matrices, eigenvalues, and eigenvectors
- Appendix C Derivatives and partial derivatives
- Index
38 - What next?
Published online by Cambridge University Press: 05 September 2012
- Frontmatter
- Contents
- Preface
- Introduction
- Part I First order differential equations
- Part II Second order linear equations with constant coefficients
- Part III Linear second order equations with variable coefficients
- Part IV Numerical methods and difference equations
- Part V Coupled 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?
- Appendix A Real and complex numbers
- Appendix B Matrices, eigenvalues, and eigenvectors
- Appendix C Derivatives and partial derivatives
- Index
Summary
In this book we have covered all of the basic methods for finding the explicit solutions of simple first and second order differential equations, along with some qualitative methods for coupled nonlinear equations. We have also discussed difference equations, and seen how complicated the dynamics of even very simple iterated nonlinear maps can become.
There are two ways in which to proceed further with the material developed here. One arises from turning first to the study of partial differential equations, while the other essentially continues from where we have left off.
Partial differential equations and boundary value problems
Partial differential equations model systems that have spatial as well as temporal structure, for example the temperature throughout an object, the vibrations of a string or a drum, or the velocity of a fluid.
In general linear partial differential equations are easier to solve. By using the technique known as ‘separation of variables’ it is possible to convert such a problem into an ordinary differential equation. This was touched on briefly in Exercise 20.10, and the exercises in this chapter apply this method in more detail for the example of the vibrating string.
- Type
- Chapter
- Information
- An Introduction to Ordinary Differential Equations , pp. 373 - 378Publisher: Cambridge University PressPrint publication year: 2004