I know this is a subjective question, but I need some opinions on a very good book for learning differential equations.
Ideally it should have a variety of problems with worked solutions and be easy to read.
I really loved Differential Equations With Applications and Historical Notes by George Simmons. It drastically changed my outlook about a large part of mathematics. For example, why do we spend so much time in real analysis studying convergence of power series? The subject is interesting on its own, but aside from the abstract interest, it’s ultimately because we want to use those methods to understand power series solutions of differential equations.
The Simmons book is clearly written, and it not only makes the subject
interesting but deeply fascinating. Great mathematicians like Gauss and Laplace were trying to solve problems of physics and engineering, in which differential equations are ubiquitous, and these problems are the primary motivation for a large part of analysis and topology. By page 30 Simmons has
treated falling objects with air resistance and shown how to calculate
terminal velocities. After spending all of high school doing falling-objects problems without air resistance, it was a relief to finally do them right.
Another early highlight is the solution of the famous brachistochrone problem, something I had been wondering for years.
Many books have a series of dry exercises of the sort “Solve the equation … using the … method.” Simmons’ exercises are juicy.
It was fun just to read them, and each one got me excited to try
to find out the answer. Here are some examples, all from chapter 1:
Consider a bead at the highest part of a circle in a vertical
plane, and let that point be joined to any lower point on the
circle by a straight wire, If the bead slides down the wire
without friction, show that it will reach the circle in the
same time regardless of the position of the lower point.
A chain 4 feet long starts with 1 foot hanging over the edge
of a table. Neglect friction, and find the time for the chain
to slide off the table.
A smooth football having the shape of a prolate spheroid 12
inches long and 6 inches thick is lying outdoors in a
rainstorm. Find the paths along which the water will run down
The clepsydra, or ancient water clock, was a bowl from which
water was allowed to escape through a small hole in the
bottom. It was often used in Greek and Roman courts to time
the speeches of lawyers, in order to keep them from talking
too much. Find the shape it should have if the water level is
to fall at a constant rate.
This one might be my favorite:
- A destroyer is hunting a submarine in a dense fog. The fog
lifts for a moment, discloses the submarine on the surface 3
miles away, and immediately descends. The speed of the
destroyer is twice that of the submarine, and it is known that
the latter will at once dive and depart at full speed in a
straight course of unknown direction. What path should the
destroyer follow to be certain of passing directly over the
submarine? Hint: Establish a polar coordinate system with the
origin at the point where the submarine was sighted.
I wouldn’t even have realized that there was such a path, and I had
to ponder for a while to persuade myself that there was.
The differential equations class I took as a youth was disappointing,
because it seemed like little more than a bag of tricks that would
work for a few equations, leaving the vast majority of interesting
problems insoluble. Simmons’ book fixed that.
[ Addendum 2020-04-20: Regarding this last point, Gian-Carlo Rota says: “The most preposterous items are found at the beginning, when the text (any text) will list a number of disconnected tricks that are passed off as useful, such as exact equations, integrating factors, homogeneous differential equations, and similarly preposterous techniques. Since it is rare – to put it gently – to find a differential equation of this kind ever occurring in engineering practice, the exercises provided along with these topics are of limited scope…”. And later: “A course taught as a bag of tricks is devoid of educational value.”. ]