Overview of Appendices
There are three appendices.
Appendix A discusses the general technique of changing
variables. Appendix B is a very brief description of the use
of power series, and Appendix C is a
short summary of some of the basic properties of complex
arithmetic.
Appendix A: Changing Variables
Many of the standard analytic techniques and many of the special cases
for analytically solving differential equations can be viewed as a
"change of variables." This appendix illustrates this
fact by discussing a number of examples. Included is a short
presentation of Bernoulli and Riccati equations.
Only the dependent variable is changed in the text and exercises
(with the exception of Exercise 22).
Comments on selected exercises
Exercises 1-13
relate solutions of equations in different variables,
both analytically and geometrically.
Exercises 14 and 15 apply this technique to mixing problems
(changing the dependent variable to concentration).
Exercise 16 introduces
homogeneous equations, i.e., equations
of the form dy/dt=g(y/t).
Exercises 17-21 illustrate
changes of variables to help with
linearization near equilibrium points.
Exercise 22 involves changing the independent variable.
Exercises 23-33 provide practice with Bernoulli and
Riccati equations.
Appendix B: The Ultimate Guess
This appendix is an ultra-lite presentation of the idea of a
power series solution to a differential equation. We
illustrate the technique with a first-order example and a
second-order example (Hermite's equation). Legendre's
equation appears in Exercise 15.
Comments on selected exercises
Exercises 1-13 illustrate the power series technique for
various first- and second-order equations. Closed form
solutions are not requested.
Exercises 14 and 15 involve Hermite's equation and Legendre's
equation respectively.
Exercises 17 and 18 provide motivation for the second
guessing technique that is discussed in Sections 1.8, 3.5,
and 3.6.
Appendix C: Complex Numbers and Euler's Formula
This appendix is not intended to teach students the arithmetic
of complex numbers, but it does summarize the basic properties that we
use, especially in Chapters 3 and 4.
There is a power series
derivation of Euler's formula and a brief mention of the polar
representation of a complex number. Both of these ideas play important
roles in Section 3.4 and Sections 4.2-4.4.