Exploring ODEs

A different kind of introduction to ODEs.

Exploring ODEs cover

Written by Nick Trefethen, Asgeir Birkisson, and myself, Exploring ODEs takes a look at ordinary differential equations unlike any other text. Rather than focusing on the mechanics of finding solutions in a limited number of special problems, we use Chebfun to illustrate the wide range of behavior of those solutions for a variety of linear, nonlinear, and multidimensional problems, including initial-value, boundary-value, and eigenvalue problems. Each short chapter includes a detailed application and a favorite reference. The text is accessible as an accompaniment or follow-up to a standard first undergraduate text on ODEs. All of the MATLAB codes generating the examples and figures are available for download, as is the full text in PDF, from the book’s website.


  1. Introduction
  2. First-order scalar linear ODEs
  3. First-order scalar nonlinear ODEs
  4. Second-order equations and damped oscillations
  5. Boundary-value problems
  6. Eigenvalues of linear BVPs
  7. Variable coefficients and adjoints
  8. Resonance
  9. Second-order equations in the phase plane
  10. Systems of equations
  11. The fundamental existence theorem
  12. Random functions and random ODEs
  13. Chaos
  14. Linear systems and linearization
  15. Stable and unstable fixed points
  16. Multiple solutions of nonlinear BVPs
  17. Bifurcation
  18. Continuation and path-following
  19. Periodic ODEs
  20. Boundary and interior layers
  21. Into the complex plane
  22. Time-dependent PDEs
  23. Appendix A: Chebfun and its ODE algorithms
  24. Appendix B: 100 more examples