Follow the department's
Math 307 syllabus.
Math 307 is required by many science and engineering majors. The
syllabus was designed after extensive consultation with these
"client" departments. Be sure to cover the required sections, and to spend
the full time allotted on applications, in particular the six lectures
allotted to the last two sections of Chapter 3.
Starting Autumn 2009, we are using the 9th edition of text by Boyce and DiPrima. There is an inexpensive "custom" version of the text containing only the chapters we use for 307 and 309. Students need either this or the full text (not both).
It is still possible, if unlikely, that a student may ask if the old (eighth) edition of the book can be used. The answer is a qualified yes. Most of the text and many of the problems are the same. Briefly, the changes for the sections in the syllabus are
You will probably be surprised by how much calculus the students don't remember. Some review problems, possibly even a quiz on review material, can be very helpful. You may be able to find examples on old course websites.
(Old course websites and other useful links are listed on the Math 307 Instructor Materials page in the Math Department Wiki.)
Suggested first class outline: Give a brief overview of differential equations (DE) and why they are useful. (Remember, you are talking mostly to future engineers.) Then do an example of an application with a separable DE. The book starts with a population problem, but some instructors find a mixing problem easier to introduce on the first day. Next discuss separable equations in general and integrating factors. Note techniques of integration that come up. (Not remembering these techniques is usually the source of most difficulties students have with the first chapter in the course.)
With this population, it's often much more effective to introduce ideas with examples and then discuss or refer to the book for the generalization. Starting a topic by giving the general case has two likely unfortunate side effects. First, many of your application-oriented students will tune out general statements filled with letters instead of numbers; examples are more likely to engage them. Second, it creates the impression the subject is just a game of memorizing formulas and plugging in constants, rather than a collection of techniques for tackling problems. If you like to use applications to make your examples more interesting, check the later exercises in each section.
Mathematica and other programs can solve all of the differential equations that are presented to the students, faster and more accurately than the students can. Teach the ideas not just how to plug and chug. The classic example is the variation of parameters method to find the general solution to a inhomogeneous equation if you are given one solution, or two solutions to the homeogeneous equation. There is a formula that results from variation of parameters, but many students misunderstand the formula, and then are completely stumped because they don't know where it comes from and hence can't go back and figure out what they are doing wrong. Note the same idea can be applied to first order, but the book doesn't teach that as a "formula".
It's important to convey a geometric and/or physical sense of what the solutions actually mean as well as formulas. (Remember, most of your students are engineers or applied scientists, not mathematicians or theoretical scientists.) The first place this comes up is using direction fields for first order equations. They are particularly helpful for explaining equilibrium solutions. Computer plotters and demonstrations may be useful; look for links to some of these on old course web pages (see Wiki page linked above). For second order equations, sketching graphs of solutions with varying coeffficients can be very helpful. A couple of instructors have borrowed a lab demonstration cart with springs from the physics department to illustrate the types of solutions for second order equations (information on Wiki page). To illustrate resonance, see Mike Munz for a video of the collapse of the Tacoma Narrows Bridge. (You may be able to find bits of this video on the web.)
Section 2.7, Euler's method, is pretty independent of the rest of the chapter. It could be covered immediately after direction fields. Or you can schedule it for shortly before a midterm, and postpone it until after the test if you need more time on other topics.
The book assumes more familiarity with complex numbers than the students have. Here is a sample handout: Notes on Complex Numbers from Bob Phelps. Do not spend a lot of time in class on arithmetic of complex numbers; students usually understand addition, subtraction, and multiplication quickly, and you can discuss division if it comes up (depending on your approach to partial fractions, see below). If you use power series to introduce the polar form, as in the sample handout, keep in mind some of your students may not have seen power series. Be sure to say the Euler Formula may be taken as a definition.
Be sure to leave enough time for Laplace transforms, as these are very important for many of the engineering students. Partial fractions need to be (re-)taught so they use this method to solve the second order equations that appear in applications. The "coverup" method is particularly simple and much less prone to errors than solving a system of equations. For irreducible quadratic factors, a modification of the standard approach can simplify computations. In place of Ax+B for the numerator, use for instance [A(x-2)+B]/((x-2)^2+9). This is helpful for two reasons: this is the form actually needed to find the inverse Laplace transform, and it is easier to find A and B as follows. Using the coverup method, multiply by (x-2)^2+9 and then let that expression tend to zero, i.e. let x tend to 2 + 3i. The real and imaginary parts give A and B separately. (For this, you will have to teach them how to divide one complex number by another.) You might be tempted to omit the problems with irreducible quadratic factors, but should not, because they occur in many of the applications.
Concerning optional sections (§§2.6, 2.8, 3.2, and 3.6), a TA teaching 307 a second time said he planned to skip most of the optional sections, "but I do plan to at least discuss the concept of a fundamental set of solutions from 3.2 because the concept pops up in a big way in Math 309. I'm also tempted to skip 3.7 [in 9th ed., 3.6], since section 3.6 [now 3.5] on its own contains an awful lot of material to absorb, and the method of undetermined coefficients can be used for all of the nonhomogeneous applications. In my experience the treatment of nonhomogeneous equations in general can gobble up an awful lot of class time because there are so many possible scenarios to illustrate, so I hope to keep it as short and sweet as possible so as not to run out of time when Laplace transforms come along."
See also the
Instructor Guide for 300 level courses.