Introduction to Ordinary Differential Equations with Mathematica®

The purpose of this companion volume to our text is to provide instructors (and eventu­ ally students) with some additional information to ease the learning process while further documenting the implementations of Mathematica and ODE. In an ideal world this volume would not be necessary, since we have systematically worked to make the text unambiguous and directly useful, by providing in the text worked examples of every technique which is discussed at the theoretical level. However, in our teaching we have found that it is helpful to have further documentation of the various solution techniques introduced in the text. The subject of differential equations is particularly well-suited to self-study, since one can always verify by hand calculation whether or not a given proposed solution is a bona­ fide solution of the differential equation and initial conditions. Accordingly, we have not reproduced the steps of the verification process in every case, rather content with the illustration of some basic cases of verification in the text. As we state there, students are strongly encouraged to verify that the proposed solution indeed satisfies the requisite equation and supplementary conditions.

Solutions.- 1. Basic Concepts.- Solutions to Section 1.1.- Solutions to Section 1.3.- 2. Using Mathematica.- Solutions to Section 2.2.- Solutions to Section 2.3.- 3. First-Order Differential Equations.- Solutions to Section 3.1.- Solutions to Section 3.2.- Solutions to Section 3.3.- Solutions to Section 3.4.- Solutions to Section 3.5.- Solutions to Section 3.6.- 4. The Package ODE.m.- Solutions to Section 4.4.- Solutions to Section 4.5.- Solutions to Section 4.4.- Solutions to Section 4.6.- Solutions to Section 4.7.- Solutions to Section 4.8.- Solutions to Section 4.9.- Solutions to Section 4.10.- Solutions to Section 4.11.- Solutions to Section 4.12.- 5. Existence and Uniqueness of Solutions of First-Order Differential Equations.- Solutions to Section 5.1.- Solutions to Section 5.2.- Solutions to Section 5.3.- Solutions to Section 5.5.- Solutions to Section 5.6.- 6. Applications of First-Order Equations I.- Solutions to Section 6.1.- Solutions to Section 6.2.- Solutions to Section 6.3.- Solutions to Section 6.4.- Solutions to Section 6.5.- Solutions to Section 6.6.- 7. Applications of First-Order Equations II.- Solutions to Section 7.1.- Solutions to Section 7.2.- Solutions to Section 7.3.- Solutions to Section 7.4.- Solutions to Section 7.5.- 8. Second-Order Linear Differential Equations.- Solutions to Section 8.1.- Solutions to Section 8.2.- Solutions to Section 8.3.- Solutions to Section 8.4.- Solutions to Section 8.5.- Solutions to Section 8.6.- Solutions to Section 8.7.- 9. Second-Order Linear Differential Equations with Constant Coefficients.- Solutions to Section 9.1.- Solutions to Section 9.2.- Solutions to Section 9.3.- Solutions to Section 9.4.- 10. Using ODE.m to Solve Second-Order Linear Differential Equations.- Solutions to Section 10.1.- Solutions to Section 10.2.- Solutions to Section 10.3.- Solutions to Section 10.4.- 11. Applications of Linear Second-Order Equations.- Solutions to Section 11.1.- Solutions to Section 11.2.- Solutions to Section 11.3.- 12. Higher-Order Linear Differential Equations.- Solutions to Section 12.1.- Solutions to Section 12.2.- Solutions to Section 12.3.- Solutions to Section 12.5.- 13. Numerical Solutions of Differential Equations.- Solutions to Section 13.1.- Solutions to Section 13.2.- Solutions to Section 13.3.- Solutions to Section 13.5.- Solutions to Section 13.6.- Solutions to Section 13.7.- Solutions to Section 13.8.- 14. The Laplace Transform.- Solutions to Section 14.1.- Solutions to Section 14.2.- Solutions to Section 14.3.- Solutions to Section 14.4.- Solutions to Section 14.5.- Solutions to Section 14.6.- Solutions to Section 14.7.- Solutions to Section 14.8.- Solutions to Section 14.9.- Solutions to Section 14.10.- 15. Systems of Linear Differential Equations.- Solutions to Section 15.1.- Solutions to Section 15.2.- Solutions to Section 15.3.- Solutions to Section 15.4.- Solutions to Section 15.5.- Solutions to Section 15.6.- Solutions to Section 15.7.- Solutions to Section 15.8.- 16. Phase Portraits of Linear Systems.- Solutions to Section 16.1.- Solutions to Section 16.2.- Solutions to Section 16.3.- 17. Stability of Nonlinear Systems.- Solutions to Section 17.1.- Solutions to Section 17.2.- Solutions to Section 17.3.- Solutions to Section 17.5.- Solutions to Section 17.6.- 18. Applications of Linear Systems.- Solutions to Section 18.1.- Solutions to Section 18.2.- Solutions to Section 18.3.- 19. Applications of Nonlinear Systems.- Solutions to Section 19.1.- Solutions to Section 19.2.- Solutions to Section 19.4.- Solutions to Section 19.5.- 20. Power Series Solutions of Second-Order Equations.- Solutions to Section 20.1.- Solutions to Section 20.3.- Solutions to Section 20.4.- 21. Frobenius Solutions of Second-Order Equations.- Solutions to Section 21.2.- Solutions to Section 21.3.- Solutions to Section 21.4.- Solutions to Section 21.5.- Solutions to Section 21.6.- Solutions to Section 21.8.- Solutions to Section 21.9.- A. Appendix: Review of Linear Algebra and Matrix Theory.- Solutions to Section A.3.- Solutions to Section A.4.- Solutions to Section A.5.- Solutions to Section A.6.- Solutions to Section A.7.- Solutions to Section A.8.- Solutions to Section A.9.