Math.321.001 Differential Equations Fall 2016
Course Objectives

Loosely speaking, the goals of the course are to learn the following:
• the important concepts, terminology, notation, and facts of differential equations
• certain types of operations, and their application to solve problems
• qualitative analysis methods for differential equations
• to write coherently and correctly about differential equation models, methods, and analyses
• how and why differential equation models are used in applications
Admittedly, parts of this list are vague.  Paradoxically, in order to understand a more specific set of objectives, you would already need to know differential equations.  The objectives would naturally mention new concepts and use terminology that you will learn in the course, but which would not make any sense to you now (unless you have already studied the subject).

To make an analogy, compare this statement of course objectives with one you might have read when first studying calculus.  For that course, a detailed and specific set of objectives would have mentioned limits, derivatives, integrals, rules of differentiation, and so on.  But until you have studied calculus the first time, these terms would have no meaning.  Just so for the present course, a specific and detailed list of the important concepts, techniques, and so on would have little meaning.

So, to flesh out the objectives, here is a little overview of the topic.  In an algebraic equation, there are unknown quantities (variables) and one of the main goals is to solve the equations to discover the values of the variables.  An answer is typically a number or a set of numbers.  In a differential equation, the unknown is a function, and a main goal is to solve the equation in order to find the function.  In the simplest of cases, that means finding an equation for the function.

Here is an example: y ' = y  is a differential equation.  It concerns an unknown function, and says that the derivative of the unknown function is actually equal to the function itself.  As you no doubt recall from calculus, ex is such a function.  So one possible solution of the differential equation is the function y =
ex.

In applications of differential equations, the unknown function often represents how some variable of interest evolves over time.  For example, the variable might represent temperature at a specific location (say the north pole).  We recognize that this is a function of time, and can write an equation like T = f (t) to represent that fact.  But that is not the same as knowing what the function IS.  With a thermometer we can record values of
f (t) as time evolves.  What is more difficult is to predict this evolution accurately in advance.  In that sense,  f (t) is an unknown function for values of time t in the future.  Finding the unknown function means being able to predict (ie, compute) its values in advance.  If we are very lucky we can find a differential equation that  models the evolution of  T  accurately, and then solve the differential equation to find the unknown function and so predict how temperature will vary in the future.

While there are many cases in which it is possible to solve a differential equation and express the answer with an equation, there are also many equations for which this is not possible.  But there are other kinds of information that we can figure out.  Does the solution increase forever?  Does it level off and approach a steady state?  Does it oscillate with a regular pattern of repetition?  Answering questions of this sort is what is meant above by qualitative analysis, a major emphasis of this course.

You may find qualitative analysis more demanding than the techniques you studied in calculus.  You will have to combine information from several different analysis tools to obtain as complete a picture as possible of how solutions to differential equations behave.  On the other hand, many students find this sort of analysis much more meaningful than solving calculus problems, and so more satisfying as well.

I hope that helps you understand the list of obectives at the top.  Please also

For another point of view on the approach taken in our course, look at A Little Philosophy and a Warning on this webpage by Robert Devaney, one of the authors of our text.