Frequently†Asked Questions
Why do we need another book on differential equations?
Good question. There are many excellent books about differential equations,
phase planes, dynamical systems and chaos. But most of them are aimed at
third year or graduate level students. This is because that is the
stage at which this material was taught to students twenty or so years ago.
But these days, many university mathematics courses include
these topics in year two, and several teach a substantial amount of it
in the first year. Therefore, there is a need for a book that covers
the essentials of this subject without including the more technical
and challenging aspects.
What are the prerequisites for this book?
You should be familiar with differential and integral calculus, and partial differentiation.
Linear algebra is also needed, especially matrices, determinants,
eigenvalues and eigenvectors. You also need to be good at sketching curves
and familiar with exponentials, logs and trig functions.
Some of these important background topics are very briefly
summarised in the appendix of the book.
Where are the solutions to the even-numbered exercises?
They are online, as supplementary electronic material, but slightly tucked away.
To find them,
first go to the Springer site for the book. Scroll down the page to
Solutions to Odd-Numbered Exercises, and click on that. Then
scroll down that page until you find
9.1 Electronic Supplementary Material, and click on the pdf file link given there.
Why does the book do discrete systems in a non-standard way?
In most books, discrete dynamical systems are called 'maps'
and are written in the form \(x_{n+1}= f(x_n)\). There
are two things I don't like about this. Firstly, the term 'map': a discrete dynamical system
is unrelated to what the word 'map' means in plain English. More importantly, when teaching this subject
over the years I have found that many students are confused that
fixed points of a map are points where \(f(x)=x\), while fixed points of a differential equation \(x'=f(x) \)
are points where \(f(x)=0\). This confusion can be avoided by writing discrete dynamical sytems
as difference equations, \(x_{n+1} - x_n = f(x_n)\), so that
fixed points of both continuous and discrete systems are points where \( f(x)=0\).
A further advantage of using this notation is that this is how discrete iterative systems arise in two
common applications - Newton's method for finding zeros of a function, and Euler's method (and most other methods)
for numerical solutions of differential equations.
† Definition: frequently - at least once.