Math 111.01/02 (Applied Mathematical Analysis I)
Fall 1998
Plan for Week 3
We start this week by studying the class of linear first-order
DEs, for which there is always a procedure for generating a symbolic
solution. As with separable equations, there is still the issue of whether
a necessary integration can be carried out in closed form. If so, a symbolic
formula for solutions can be found. This week's lab is a further exploration of first-order linear equations -- in particular, of the significance of the coefficients in these equations.
At midweek we explore what might go wrong if the DE is nonlinear.
We will see that there can be initial value problems that have either no
solution or more than one solution -- but these are very rare and
not likely to occur in practice. In particular, if dy/dt = f(t,y),
where both f and fy are continuous in some region
of the (t,y)-plane, then every initial value problem that starts
in that region will have a unique solution through the starting point.
This has important consequences that we can count on, even if we can't
find formulas for the solutions.
Here is the syllabus for Week 3:
| Week 3 |
Date |
Topic |
Reading |
Activity |
|
|
|
|
|
| M |
9/14 |
First-order linear DEs |
2.1-2.2 |
|
| W |
9/16 |
Qualitative behavior |
2.4 |
Worksheet: First order linear DE's |
| F |
9/18 |
First-order linear DEs |
|
Lab: World Class Sprints |
|
|
|
|
|
Notes
- Your next homework papers will be turned in on Monday, September 21.
Those papers should include solutions to all problems in the assignment
below. The assignment dates are start dates.
- Your textbook has far too many answers in the back of the book. If
you look there first, you will subvert the learning process. If you know
your answers are correct, you will never need to look there. In general,
no homework problem will be given full credit unless you have written
an explanation of why you know it is correct. (Exceptions to
this rule are the exercises whose numbers appear in parentheses. If the
problem starts with "show", the answer and the explanation may
be essentially the same thing.) For example, an acceptable explanation
for a solution of a differential equation is that you have substituted
the proposed solution into the original equation and found that it satisfies
the equation -- but you have to show your work. Two examples of unacceptable
explanations: (a) It matches the answer in the back of the book. (b)
I did the work again and it came out the same.
- Submit your Week 3 lab report (the Maple file) via e-mail by the end
of the day Wednesday, September 23.
- Remember to submit your e-mail journal entry on Friday, September 18.
- Your instructor will be going out of town after class on Friday this week and will return before the following Monday's class.
Assignments
- Monday, Sep. 14, Sec. 2.1 / #1, 4, 10, 13, 23; Sec. 2.2 / #13, 25
- Wednesday, Sep. 16, Sec. 2.4 / #1, 4, 9, 13, 14
- Friday, Sep. 18, Sec. 2.1 / #17, 22; Sec. 2.4 / #17
David A. Smith <das@math.duke.edu>
Last modified: September 12, 1998