Math.321.001 Differential Equations Fall 2016
Is this the right course for you?

As explained in the page on Course Objectives, this course will emphasize qualitative analysis of differential equations.  This is a fairly modern approach to the topic, and will differ from traditional calculus courses in important ways.  Please read further for more details about the qualitative approach and why I prefer it.  After reading the rest of the page, if you think you would prefer a differential equations course with a more traditional emphasis, please consider transferring to the other section, meeting 9:45 - 11 AM MTh.

In describing the qualitative approach, I will quote from th
is webpage posted by Robert Devaney, one of the authors of our textbook.  He says:

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 ex = x + 5 cannot be solved using algebra.  There is no way to combine the familiar operations and functions of calculus to obtain an exact solution.    By trial and error (or more sophisticated methods) we can find an approximate solution is given by
x = -4.99321618864790....   Something similar occurs with differential equations, but in that context the unknown is a function, not a number.  For example, consider the differential equation y' y = 2et.   Finding a solution means finding a function for which multiplying the function by its own derivative results in 2et.  One possibility is y = 2et/2  That is an explicit solution.  In contrast, the differential equation  y' y = 2e ^ t2  (that is e with an exponent of  t2 )  does not have an explicit solution: there is no familiar calculus function that can be substituted for y to make the equation true.

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.

The last sentence is key.  Devaney is a nationally recognized expert on differential equations and applications.  He is telling us here that the traditional differential equations course will be more comfortable to many students, but may not be terribly relevant to actual applications students will encounter in their careers.   Of course, some students are not motivated by applications, and enjoy learning about math for its own sake.  From that perspective, there is a lot of interesting material in a traditional differential equations course.  My point here is not to hold up one approach or the other as better.  Rather, I want students to be aware of the different approaches that can be taken, and to make an informed decision about which course best suits their needs and objectives.