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
- 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
- Appendix A Real and complex numbers
- Appendix B Matrices, eigenvalues, and eigenvectors
- Appendix C Derivatives and partial derivatives
- Index
28 - Distinct real eigenvalues
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
- 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
- Appendix A Real and complex numbers
- Appendix B Matrices, eigenvalues, and eigenvectors
- Appendix C Derivatives and partial derivatives
- Index
Summary
In this chapter, and the following two, our main aim is to show how to use a knowledge of the eigenvalues and eigenvectors of in order to draw the phase diagram for the equation ẋ =x. As in Chapter 7, this phase diagram will illustrate the qualitative behaviour of the solutions by showing a representative choice of the curves traced out by the solutions (x(t), y(t)), labelled with an arrow to indicate in which direction the solution moves as t increases.
In each chapter we will examine one of the three possibilities (two distinct real eigenvalues, a complex conjugate pair of eigenvalues, or a repeated eigenvalue) and for each case we will show
(i) how an appropriate change of coordinates, based on the eigenvectors of the matrix, can be used to transform the differential equation into a standard, simpler (canonical) form;
(ii) how to find the explicit solution of this simple form of the equation;
(iii) how to draw the phase portrait for the simple equation;
and hence
(iv) how to find the explicit solution of the original equation; and
(v) how to draw its phase portrait.
Since we already have a reliable method for solving coupled linear equations, (ii) and (iv) will be much less important than (iii) and (v).
- Type
- Chapter
- Information
- An Introduction to Ordinary Differential Equations , pp. 269 - 284Publisher: Cambridge University PressPrint publication year: 2004