Math.321.001 Differential Equations Fall 2016

**Is this the right course for you?
**

In describing the qualitative approach, I will quote from th

This
is a course in ordinary differential equations. However, the course is
by no means "ordinary." In years past, before we had widespread access
to computers and computer graphics,
courses in ordinary differential equations consisted mainly in a
series of special tricks to solve special differential equations.

The tricks Devaney is referring to here are similar to methods of integration studied in calc 2. But the need for students to master a large tnumber of techniques in either of these areas has greatly diminished since the advent of computer algebra systems (such as Mathematica or WolframAlpha), that can apply the techniques faster and more accurately than a human working problems out by hand. If an application involves differential equations that can be solved using traditional methods, it is usually better to rely on software. Devaney says a modern course in differential equations should focus instead on differential equations that can NOT be solved by the traditional methods. Continuing to quote him:

Unfortunately,
most differential equations (and in particular most
differential equations that arise in applications) cannot be solved
explicitly by these or any other method.

What does solving explicitly mean? It refers to finding an equation for a function f (t) that satisfies the given differential equation. To use an analogy, the equation e

Devaney continues:

Today
we all have access to high-speed computers and computer graphics. Like
humans, computers cannot solve most differential equations that arise.
However, they can give us an APPROXIMATE or NUMERICAL solution. For
many purposes, this is good enough.

Unfortunately, computers make mistakes (sometimes because of round-off errors or sometimes because the differential equation is not suited to numerical approximation). So we always have to be careful when we solve differential equations this way. More importantly, the output of the computer is not a formula that we can use to compute values of our solutions. Rather, the output from the computer is a rather lengthy list of numbers. Most often, it is best to view this list geometrically as a phase line or plane or other geometric object.

All of this means that this will be a very different type of mathematics course. In this course you will rarely be asked to generate specific formulas for solutions of differential equations. Rather, you will be asked to understand the algorithms that lead to numerical solutions, to interpret the resulting pictures produced by the computer, and to relate these images back to the original application.

.... Your homework problems and questions on exams will often involve essays rather than simple routine computations. And you will often have to use technology to come up with answers to questions that are posed. ... Most students in the past have found this kind of course quite challenging, but lots of fun. If you are used to the old style of mathematics courses, be prepared for something quite different and perhaps much more relevant to whatever your use for differential equations is.

Unfortunately, computers make mistakes (sometimes because of round-off errors or sometimes because the differential equation is not suited to numerical approximation). So we always have to be careful when we solve differential equations this way. More importantly, the output of the computer is not a formula that we can use to compute values of our solutions. Rather, the output from the computer is a rather lengthy list of numbers. Most often, it is best to view this list geometrically as a phase line or plane or other geometric object.

All of this means that this will be a very different type of mathematics course. In this course you will rarely be asked to generate specific formulas for solutions of differential equations. Rather, you will be asked to understand the algorithms that lead to numerical solutions, to interpret the resulting pictures produced by the computer, and to relate these images back to the original application.

.... Your homework problems and questions on exams will often involve essays rather than simple routine computations. And you will often have to use technology to come up with answers to questions that are posed. ... Most students in the past have found this kind of course quite challenging, but lots of fun. If you are used to the old style of mathematics courses, be prepared for something quite different and perhaps much more relevant to whatever your use for differential equations is.